B 

0 
0 
0: 

01 
1  ^ 
21 

9[ 
4  I 
3  1 


TUE 


POCKET  REFERENCE 


ED  ^S  A  WOEK    OF   FEACTICAL  UTILITY 
TEVCFTEK,    THE   MAN   OF  BUSINESS,  AN1> 
HX3IAX10AL  flTUBJSKT. 


By  I.  M.  WILCOXSON; 


HUKK   SfRlIfGS,   HART  COrNTT,  KBNT*         T 


L0UI8VII.liB,  KY: 

MORTO.V    &   GRISWOt.D,    PrYtE'S 

1858. 


ARITHMETICAL  VADE  MECUM; 


POCKET    REFERENCE- 


DieiGNED  AS  A  WORK  OF  PRACTICAL  UTILITY,  FOl 

THE  TEACHER,  THE  MAN  OF  BUSINESS, 

AND  THE  MATHEMATICAL 

STUDENT. 


By  I.  N.  WILCOXSON, 

THEEE  SPRINGS,  HABT  COUNTY,   XT. 


LOUISVILLE : 

A.  F.  OOX,    STBBBJEOTTPER    AND    rSINria. 

1856, 


Entered  according  to  act  of  Congress,  in  the  yeai  1856,  by 

I.    N.    WILCOXEN, 

In  the  Clerk's  Office  for  the  District  of  Kentucky 


sRie 

URU 


TO 

REV.  J.  P.  MURREL,  i 

rBINCIPAL     OF     CAUDEN     SEMIKAET, 

IN  TOKEN 

OF  OCR   HTOn   APPRECIATION   OF   HIS   LIFE   AND   CHARACTBB 

AS  A 

TEACHER    OF   YOUTH; 

AND  TO 

ED.  PORTER  THOMPSON, 

MT    YOU.VQ    FRIEND     AND    FELLO^-STCDENT, 
THIS   LITTLE    WORK  IS 

BESrECTFULLY    AND 

FRATERNALLY 

INSCRIBED, 

BY   THE 

AUTHOR, 


PREFACE. 

It  may  be  thought  that  the  great  number 
of  Arithmetical  works,  now  before  the  public, 
renders  it  unnecessary  to  multiply  them  ;  but  in 
adding  another  to  the  list,  we  deem  it  sufficient 
apology  to  remark,  that  the  importance  of 
mathematical  science  renders  it  justifiable  in  any 
one  to  present  to  the  public  every  improvement 
that  will  enable  calculators  to  arrive,  with  greater 
facility,  from  premises  to  conclusions.  The 
increasing  interest  manifested  in  the  cause  of 
Education,  and  the  labor  of  Rev.  J.  P.  Murrel, 
Principal  of  Camden  Seminary,  together  with 
that  of  the  author  of  these  pages,  for  many 
years,  in  abridging  and  simplifying  the  common 
methods  of  calculation,  are  sufficient  inducements 
for  the  publication  of  the  result  of  those  efforts. 
The  method  of  statement  given,  is  better  adapted 
to  all  questions  that  may  arise,  in  business 
transactions,  than  any  heretofore  offered. 

We  have  arranged  the  common  rules  of 
Practical  Arithmetic  —  Simple  and  Compound 
Proportion,  Practice,  Interest,  Discount,  Barter, 
Percentage,  and  those  for  all  kinds  of  Mensura- 
tion —  under  the  general  head  of  Cause  ana 
Effect ;  and  it  will  be  perceived  that  the  method 


▼1  PREFACE 

of  reasoning  cmploj-ed,  adapts  itself  to  the 
most  common  capacity,  and  is  applicable  to 
the  solution  of  all  c^uestions  that  may  arise 
under  any  of  those  individual  heads.  As  the 
work  is  designed  for  the  teacher,  the  business 
man,  and  the  advanced  mathematical  student, 
and  as  our  object  has  been  to  illustrate  the 
nature  and  principles  of  calculation,  we  have 
given  only  a  limited  number  of  problems ; 
deeming  it  unnecessary  to  give  more  than  a 
sufficiency  to  illustrate  the  rules,  as  competent 
teachers  can  readily  give  examples  under  every 
head,  and  to  the  business  man  questions  will 
naturally  arise. 

It  will  be  observed  that  we  have  mostly  used 
abstract  numbers ;  but  as  it  is  presumed  that 
those  who  use  the  work  will  have  previously 
become  acquainted  with  both  the  Simple  and 
Compound  rules  of  Addition,  Subtraction,  Mul- 
tiplication, and  Division,  they  can  adapt  the 
rules  here  given  to  the  solution  of  all  questions 
involving  different  denominations. 

Indulging  the  fond  hope  that  this  little  work 
may  prove  itself  worthy  the  consideration  of 
those  for  whom  it  is  designed,  we  now  submit 
it  to  a  generous  public. 

I.  N.  WILCOXSON. 

Ifnrt   County,  Kentucky,  August,  1855. 


INDEX. 


Dbdication—          .... 

.     Poge  I 

Preface—          .           .           .           ... 

i 

Explanatory  Articles —  .           .           . 

9 

Explanation  of  Signs,          .            .            . 

.        10 

Cancellation —       .... 

10 

Vulgar  Fractions — 

12 

Examples  of  Different  Kinds  of  Fractions, 

12 

Reduction  of  Vulgar  Fractions,      . 

12 

Addition  of  Vulgar  Fractions,  . 

13 

Subtraction  of  Vulgar  Fractions,   . 

.        14 

Multiplication  of  Vulgar  Fractions, 

14 

Division  of  Vulgar  Fractions, 

.        14 

Reduction  of  Complex  Fractions,          . 

U 

pEciMAL  Fractions — 

.        16 

Numeration  Table, 

18 

Addition  and  Subtraction  of  Decimals, 

.        17 

Multiplication  of  Decimals, 

17 

Division  of  Decimals, 

.        17 

Reduction  of  Decimals, 

IS 

Factor  Tables—         .... 

.        18 

Cause  and  Effect— 

20 

Simple  Proportion, 

.        20 

Definition  of  Terms, 

21 

Compound  Proportion,         .            .            . 

.        23 

Practice,              .... 

26 

Interest,                    .... 

26 

Interest  Table,    .... 

27 

Miscellaneous  Examples,     .            .            . 

29 

Percentage, 

29 

Barter,          ..... 

32 

Discount,             .            .            .            .           . 

33 

Measurement           .           .            .         ,     . 

Zi 

ym 


INDEX. 


Mensuration  Table, 

.  Pagi 

3S 

Cord  Wood, 

35 

Land  Measure, 

36 

Carpeting,     . 

37 

Crib  and  Box  Measure, 

zr 

House  Covering, 

39 

Brick  Work, 

41 

Log  and  Plank  Measure,      . 

42 

Promiscuous  Examples, 

44 

Mknscbation  of  Superfices— 

45 

Mensuration  op  Solids- 

50 

Square  Root- 

> 

52 

Cube  Boot— 

63 

Table  of  Squares  and  Cubes, 

. 

55 

Application  of  Square  and  Cube  Root, 

with  Mia- 

cellaneocs  Matter— 

, 

, 

&7 

FalliDg  Bodies, 

• 

•           • 

M 

EXPLANATORY  ARTICLES. 


Article  1.  In  placing  numbers  on  the  line, 
place  the  individual  numbers  directly  under  each 
other.  If  fractions,  place  the  numerator  where 
you  would  place  it  were  it  a  whole  number,  with 
its  denominator  on  the  opposite  side.  The  solu- 
tion of  fractional  sums  in  Multiplication  and 
Division  will  then  be  nothing  more  than  those  in 
whole  numbers. 

Art.  2.  Observe  that  we  cannot  add,  sub- 
tract, multiply,  or  divide  numbers  of  different 
denominations,  without  first  reducing  them  to 
the  same  denomination  ;  also,  that  no  ratio  can 
exist  between  different  denominations,  but  the 
abstract  numbers  rej)resenting  them  may  have  a 
ratio. 

Art.  3.  To  render  verification  easy,  observe 
that  subtraction  is  the  converse  of  addition,  and 
division  of  multiplication. 

Art.  4.  A  unit  is  the  basis  of  all  work  ;  the 
point  from  whence  the  analytical  student  reckons 
with  facility  ;  for  the  value  of  a  fraction  is  de- 
creased when  multiplied  by  itself,  while  any 
number  greater  than  one  is  increased.  Unity  is 
the  result  of  any  fraction  divided  by  itself ;  and, 
1-^-1=1.     1x1=1. 

Art.  5.  Cancel  as  much  as  possible  in  every 
Instance,  as  it  will  aflord  every  opportunity  of 


10 


ARITHMETICAL 


shortening  the  work,  and  in  no  case  of  length- 
ening  it. 

Explanation  of  Signs. 

The  sign  =  equality,  thus,  100  cents=$l 

♦*     +  addition,  "  3-f  3=6 

"     — ■  subtraction,        *'  6—3=3 

"       '*     X  multiplication,  "  3x3=9 

"       "     -1.  division,  *'  9-7-3=3 

or  thus,  I       =3 

319   =3 

"       "  ::::  proportion,    "  2:4::3:6 

Read  2  is  to  4,  as  3  is  to  6. 
**       "      J    square  root,  or  radical, 
showing  that  the  square 
root  is  required,  thus,      ^9=3 

"       **  vinculum,  showing  that 

all  under  it  is  taken  as 

one  number, 

thus,  ^100— ]  9=9 
The  exponent  (^  ),  shows  that  the  num- 
ber over  which  it  is  placed  must  be 
squared,  thus,  6*  xk36 


CANCELLATION. 


Art.  6.  First  draw  a  perpendicular  line,  which 
in  all  instances  separates  the  dividend  from  the 
divisor,  or  the  factors  of  the  dividend  from  those 


VAUE    MECLM. 


n 


of  the  divisor.  The  dividend,  or  its  factors,  is 
always  placed  on  the  right ;  the  divisor,  or  its 
factors,  on  the  left. 

Art.  7.  After  having  placed  the  sum  on  the 
line,  divide  the  continued  product  on  the  right, 
by  the  continued  product  on  the  left. 

Art.  8.  Much  mechanical  labor  may  be  saved, 

1st,  By  canceling  all  equal  numbei-s  of  noughts 
on  opposite  sides  of  the  line. 

2d,  By  canceling  all  equal  numbers  of  figures. 

3d,  If  the  product  of  any  numbers  on  one  side 
will  equal  any  number,  or  the  product  of  any 
numbers,  on  the  opposite  side,  by  canceling  all 
such  numbers. 

4th,  By  canceling  all  numbers  that  are  divisi- 
ble one  by  the  other,  *  placing  the  quotient  on 
the  side  of  the  greater. 

5tli,  By  continuing  to  cancel  any  numbers, 
(on  opposite  sides  of  the  line),  that  are  divisible 
by  any  supposed  number,  using  the  quotients. 

6t]i,  The  operation  may  be  completed  by  (Ar- 
ticle 7). 

^^ote.  The  answer  always  comes  on  the  right. 
If  the  divisor  is  greater  than  the  dividend,  the 
answer  will  be  a  fraction. 

y.  B.  The  preceding  should  be  committed  to 
memory,  and  applied  throughout  the  work. 

*The  divisions  on  tho  lino  must  all  be  made  without* 
remainder,  until  the  last. 


12  ARITHMETICAL 

VULGAR  FRACTIONS. 


Art.  9.  A  fraction  is  a  part  of  a  unit.  The, 
bottom  is  the  divisor,  or  denominator,  showing 
into  how  many  parts  a  unit  is  divided.  The  top 
is  the  dividend,  or  numerator,  showing  how 
many  of  such  parts  are  taken. 

Examples  of  the  different  hinds  of  Fractions, 

i-,  |,  I,  etc.,  Proper  fractions. 
r>  if  h  6tc.,  Improper  fractions. 

I,  |,   !■  Complex  fractions. 

3        8) 

12^,  6|,  etc.,  Mixed  numbers. 

Art.  10.  1.  Reduce  6§  to  an  improper  fraction. 
6x3-|-2=20,  numerator, 


Ans.%'. 


3,  denominator. 
2.  Reduce  12\  to  an  improper  fraction. 

IS.  '2 
Art.  11.  1.  Reduce  V  *^  ^  mixed  number. 

75h-4=18|. 
2.  Reduce  V  ^^  ^  mixed  number.     Ans.  7^ 
Art.  12.  1.  Required  to  reduce  y/o  to  its  low 
est  terms. 

(See  Article  8.)       -^\^-YAns. 
2.  Reduce  ^V  ^^  ^^s  lowest  terms.     Ans.  §. 


YADIS    UECUM. 


18 


Art.  13.  To  find  the  least  common  denomi- 
nator of  fractions. 

Place  to  the  left  all  the  prime  numbers  of  the 
denominators  and  the  least  number  of  prime  factors 
that  with  the  prime  numbers,  any  two  or  more  of 
them  multiplied  together  will  equal  any  of  the  com- 
posite denominators,  their  continued  product  will  be 
the  least  common  denominator, 

1.  Required  to  find  the  least  common  denomi- 


nator of  \,  f,  \,  ^, 


Divisors.  •< 


211  =  1 

812=1 

4il  =  l  {'Quotients. 

61=1 

8l3=l 


2.  Reduce  | 


Com.  denom.  24 

8'  I'  h  To.  to  a  common  denom- 
Ans.  120. 


Addition  of  Vulgar  Fractions. 


inator. 
Art.  14 

Rule.  Find  the  least  common  denominator. 
For  the  numerators  divide  the  common  denomina- 
tor by  each  particidar  denominator,  and  multiply 
the  quotient  by  its  numerator. 


1.  Add  I, 

h  t\,  ' 

-V,  and  |.  together. 

6:  6 

5=  1 

"50^ 

2:  2 

1=  1 

30 

5  :10 

9=  1 

54  ^new  numerators. 

:12 

11=  1 

55 
24 

:  5 

2=  1 

60i213=3H  Ans, 


14  ARITHMETICAL 

2.  Add  I,  L,  I,  I  and  |  togetlier.     Ans.  3||. 
Art.  15.   Subtraction  of  Vulgar  Fractions. 

Hule.     Prepare  the  fraction  as  in  Addition, 
and  take  the  difference  of  the  numerators. 
1.  From  §,  take  y^- 


4:  4 

3=  1,9 

:12 

5=  3|5 

12|4=i.  Ans. 

2.  From  4,  take  fV.     Ans,  \. 

Art.  16.  Multiplication  of  Vulgar  Fractions. 

Rule.     Place  all  the  numerators  on  the  right. 

QTid  the  denominators  opposite. 

1.  Multiply  ^  of  1  of  1  of  f  of  1  of  I,  by  14. 

A? 

m 

?l^ 

m 

1 

1  2  Ans. 

2.  Multiply  i  of  yV  of  VI  of  1,  by  y\  of  f  of 

12.     Ans.\. 

Note.  Of,  between  fractions  always  denotes 
multiplication. 

Art.  17.  Division  of  Vulgar  Fractions. 

Rule.  Place  the  numerators  of  the  dividend 
on  the  right,  and  the  numerators  of  the  divisor 
on  the  left.     (See  Art.  1.) 


VADE    MECDM. 


15 


1.  Divide  ^  of  f  of  Jy  of  H»  ^7  /a  of  \  of  j\. 


t\x% 

1|4 

I  4  -4n5. 

2.  Divide  \  of  ^i  of  |  of  ^f ,  by  i-  of  y\.  ^rw.  4. 

JVb/e.  To  understand  multiplication  and  divi- 
iion  of  fractions,  it  is  sufficient  to  observe  Art, 
9,  6  and  8. 

Art.  18.    Reduction  of  Complex  Fractions. 

Rule.  Consider  the  numerator  the  dividend 
and  the  denominator  the  divisor,  and  proceed  as 
in  Division  of  Fractions. 


numerator. 


1    - 

f   denominator. 

4  5 

f  of  -?- 


2.   Reduce 


This  is  the  same  as  |-r-|. 
1 0  oi  — y  to  a  mixed  number. 

^2 


12 

1 

XX 


4 
3 

X^ 
xp  11 

33 

8^  Ant. 


16  ARITHMETICAL 

3.   Kfiduce   t\  of  |  of  y|.    Ans.  2\, 


DECIMAL   FRACTIONS. 


Art.  19.  Decimal  Fractions  are  managed  like 
whole  numbers ;  and  as  their  denominators  art 
always  10,  100,  etc.,  as  y%-,  yV^,  etc.,  we  express 
them  by  placing  a  decimal  point  to  the  left  of 
the  numerator  ;  thus,  .2,  .25,  etc.  Noughts  to 
the  left  decrease  their  value  in  a  tenfold  ratio. 

Numeration  Table, 


ii    el  i        .11    il 

4444444444     •      444444444 
Ascending,  Descending. 

Kote.  This  table  shows  ''Jiat  the  value  of 
figures  is  determined  by  their  distance  from  th« 
decimal  point. 


VADE    MECDM.  17 

Art.  20.  Addition  and  subtractio  i  of  Decimals. 

Rule.  Place  the  decimal  points  directly  under 
each  other,  and  add,  or  subtract,  as  in  whole 
numbers. 

Add  .4,  .06,  1.12,  10.002,  together. 

.4 

.06 

1.12 

10.002 


11.582  Ans. 
From  3.856,  take  2.412.         3.856 

2.412 


1.444    Ans. 
Art.  21.  Multiplication  of  Decimals. 

Hule.  Multiply  as  in  whole  numbers,  and 
point  off  in  the  product  as  many  figures  for 
decimals  as  there  are  decimal  places  in  both  factors. 

Note.  Point  off  from  the  right,  and  if  thera 
are  not  enough  figures  in  the  product  to  supply 
the  decimal  place,  prefix  noughts. 

1.  Multiply  2.34  by  .12.  2.34 

.12 


.2808 
2.  Multiply  .275  by  .25.  Ans.  .06875. 

Art.  22.  Division  of  Decimals. 

Rule.  Make  an  equal  number  of  decimal  placet 
in  both  factors,  by  annexing  ciphers  to  eiiher,  and 
divide  as  in  whole  numbers. 

2 


18 


ARITHMETICAL 


Note.  The  answer  will  be  in  whole  nnmbers. 
If  decimals  are  required,  annex  cijihers  to  tl^ 
remainder,  and  continue  the  division. 

Divide  2.3421  by  21.1. 

21.100012.3421 


10.111  Ans. 

Art.  23.    1.  What  decimal  is  equivalent  to  f  ? 
4|3.00 

1  .75  Ans. 
2.  What  decimal  is  equivalent  to  |-?  Ans.  .125. 

Art.  24.    1.  Reduce  .75  to  a  vulgar  fraction. 
4  i'.pj31^  3 


I  Ans. 


2.  Heduce  .125  to  a  vulgar  fraction.     A3iu,\. 


FACTOR  TABLES. 


Federal  Money. 

10  mills  (•.) 

make  1  cent. 

marked 

et. 

100  ceots 

1  dollar. 
Avoirdupoin  Weijjht. 

$ 

16  drama  {dr.) 

make  1  ounce, 

marked 

oz. 

J  6  ounces 

"       1  pound, 

" 

lb. 

25  pounds 

"       1  quarter, 

ti 

qr. 

4  quarters 

"      1  buudrud 

weight,  " 

ewU 

20  hundred 

«      1  ton, 

<« 

r. 

VADE    MECCM.  19 

Ling  Meaaure, 

12  inches  (in.)               maku  1  foot,                marked  ft. 

3  feet                                  "      1  yard,                        "  yd. 
5^  yards                          "      1  pole  or  perch,        "  p. 

40  poles                                "      1  furlong,                   "  fur. 

8  furlongs                        «      1  mile,                      "  J/. 

Land,  or  S({uare  Measure. 

144  square  inche3(  « J.  ih)  make  1  square  foot,    marked  'q.ft. 

9  square  foct  "  1  square  yard,  "  aq.  n'l. 
'H)]4  square  yards  "  1  square  pole,  "  p. 
40  square  y<Acs                   "      1  rood,                        "  Jl. 

4  ro(jd3                                 "       1  acre,                         "  A. 

S'Jid,  or  Cubic  JL^surc. 

172*^  srlM  inches  (s.in)  m.ike  1  solid  foot,        marked  t.ft. 

27  solid  feet,                     *'      1  solid  yard,              *'  ».  yd. 

US  solid  feet,  SX4X4       "      1  cord  of  wood,        «  C. 

13^  solid  feet,                 «      1  bushel,                  «  bu. 

XoTE.     1^  solid  feet  make    1  bushel  shelled  ^rain,  two 
measures  are  given  when  corn  in  the  ear,  (2)^  solid  feet.) 

0  bushels  make  1  barr.l. 

Time. 


60  seconds 

{sec 

) 

make 

1  minnte, 

marked 

min. 

fiO  minutes 

'• 

1  i..ur. 

'• 

h. 

24  b..urs 

« 

1  dny. 

u 

d. 

30  days 

a 

1  month, 

" 

tn. 

12  mJnths 

« 

1  year. 

« 

y- 

Note.  The  limits  of  this  work  will  not  admit  of  superflu- 
ous matter,  hence  we  have  only  used  the  factors  tliat  are 
necessary  in  t'hia  work,  as  others  may  be  found  in  every  work 
extant.  In  all  practica  work,  30  days  arc  considered  a  month, 
ftnd  we  have  given  the  above  table  aa  a  rofcreuco  fur  pracu- 
oal  purposes. 


20  ARITHMETICAL 


CAUSE  AND  EFFECT. 


Art.  25.  The  older  method  of  stating  ques- 
tions by  Simple  and  Compound  Proportion,  and 
other  rules,  is  very  good  as  a  mechanical  con- 
trivance ;  hut  the  following  is  preferable,  because 
of  its  greater  simplicity  and  more  extended  ap- 
plicability. It  is  an  axiom  in  philosophy,  that 
equal  causes  produce  equal  effects,  and  that  effects 
are  always  proportionate  to  their  causes.  This 
principle  gives  rise  to  a  rule  applicable  to  all 
questions  that  can  arise  under  proportion. 

Rule.  Any  given  cause  is  to  its  effect,  as  any 
required  cause  *  is  to  its  effect ;  or,  as  another 
given  cause  to  its  required  effect.  Or,  as  any 
cause  is  to  another  similar  cause,  so  is  the  effect 
of  the  first  cause  to  the  effect  of  the  second. 

Previous  to  giving  examples  illustrative  of  the 
rule,  we  will  give  the  definitions  of  a  few  terms 
a^  used  in  this  work ;  and  then  merely  mention 
the  heads  of  rules  as  denominated  in  previous 
works,  treating  of  their  principles  exclusively 
under  cause  and  effect. 

Simple  Proportion. 

Art.  26.  1.  The  quotient  of  one  number  divi- 
ded by   another  is   called  .the   ratio.      2.  Two 

♦Causes  in  thosamo  proportion  must  always  be  of  the 
iame  kind,  and  must  bo  reduced  to  the  same  denomination 
before  placed  on  the  line. 


VADE   MECUM.  21 

numbers  compared  are  called  a  couplit.     3.  The 

first  and  second  terms  of  a  proportion  constitute 
the  first  couplet.  4.  The  third  and  fourth  terms 
constitute  the  second  couplet.  5.  The  first  and 
fourth  terms  are  called  the  extremes.  6.  The 
second  and  third  terms,  the  means.  7.  A  pro- 
portion is  an  equality  of  ratios. 

Art.  27.  And  since  each  effect,  divided  by  its 
cause,  or  each  cause,  dirided  by  its  effect,  produces 
equal  ratios,  it  follows  that  the  product  of  the 
means  must  equal  the  product  of  the  extremes. 
This  is  true  in  every  proportion,  otherwise  it  is 
not  a  proportion. 

Take  the  proportion  —  2  :  4  ::  3  :  6,  2x6= 
4X3. 

Art.  28.  The  terms  of  any  proportion  may  be 
changed  eight  different  ways,  and  still  constitute 
a  proportion  ;  but  the  ratios  will  be  ditferent. 
Again,  we  may  multiply  or  divide  two  propor- 
tions, term  by  term,  and  the  result  will  be  a 
proportion,  etc.  If  we  treat  both  couplets 
exactly  alike,  no  matter  what  we  do,  the  result 
will  be  a  proportion,  and  the  product  of  the 
means  will  equal  the  product  of  the  extremes. 

Art.  29.  By  retaining  strong  hold  on  this  fact, 
we  may  find  any  lost  term  or  factor  of  a  term. 

Take  the  proportion — 2  :  4  ::  5  :  10. 

Let  the  fourth  term  be  wanting,  or,  as  we 
shall  denominate  it,  blank,  as  2  :  4  ::  5  :  [], 


* 


*  We  shall  use  brackets  to  denote  the  place  of  the  lost,  or 
required  term,  or  factor. 


22 


ARITHMETICAL 


We  still  have  the  means,  and  one  factor  of  the 
extremes.  The  means  are  complete,  and  the  ex- 
tremes incomplete ;  hence,  in  placing  them  on 
the  line  proceed  thus  : 

Place  the  complete  on  the  right,  and  the 
incomplete  on  the  left ;  or,  make  a  dividend  of 
ilie  complete,  and  a  divisor  of  the  incomplete. 

Complete  the  operation  by  Art.  6,  7  and  8. 

^ote.     In  performing  operations 

on  the  slate  or  blackboard,  instead  ^42 

of  placing  the  quotients  to  the  right  5 

and  left  of  their  individual  dividends, 

it  will  be  more  convtnient  to  place  10  Ans 
them  directly  under  them. 

Let  the  second  term  be  blank  —  2  :  []  ::  5  :  10 

2 


4  Ans. 

Let  the  third  term  be  blank 
X^  5 


2:4::[]:10. 


M 


^ 


5  Ans. 


Let  the  first  term  be  blank  —  []  :  4  ::  5  :  10. 

$  X^\i  ^ 


\2  Ans. 

Art.  30.  The  first  caution  is  to  strictly  exam- 
ine numbers,  in  doing  which  it  will  be  seen  that^ 


TADE    MECDM.  2f<> 

in  cause  and  efect,  the  actual  numbers  given  need 
not  be  used,  provided  we  use  their  proportionate 
numbers. 

If  48  lbs.  of  pork  cost  144  cts.,  what  will  115 
lbs.  cost  ? 

Cause.  Effect.        Cause.    Effect. 

48  :  144 


1  :       3 


wv&m 


This  is  simply  another  fonn  of  Art.  8.  It  ia 
readily  seen  that  1  :  3,  is  as  48  :  144. 

Compound  Proportion. 

Art.  31.  In  this  system  of  statement,  the 
philosophical  idea  is  the  only  sure  guide  ;  hence, 
in  stating  a  question,  the  mind  should  rest  on 
denominations  only  ;  but  after  it  is  stated,  we 
should  look  on  the  terms  as  abstract  numbers. 

JV.  B.  \Yhen  a  correct  statement  is  made, 
there  will  be  the  same  number  of  elements,  or 
factors,  under  the  same  letters,  or  in  similar 
terms  :  as  in  the  following  example.  If  2  men 
in  4  days  can  mow  5  acres  of  grass  by  working 
10  hours  per  day,  how  many  acres  will  6  men 
mow  in  2  days  by  working*  12  hours  per  day  ? 

An^.  9  A. 

Cause.  Effect.  Cause.  Effect. 

Men,       2  :  Acres,  b::Men,       6  :  Acres  [] 
Days,     4 :  Days,     2 : 

Ifours,  10  :  Hours,  12  : 


24  ARITHMETICAL 

4  \x^  ^    • 


$  M^ 


3 


I  9  Ans. 

Art.  32.  There  are  many  questions  that  appeaj 
to  be  in  simple  proportion,  that  are  really  in 
compound  proportion  ;  the  reason  is,  because  of 
one  terra  in  each  couplet  being  the  same. 

Example — If  4  men  build  a  wall  in  8  days, 
6ow  long  will  it  require  6  men  to  build  it  ? 

Cause.  Effect.  Cause.  Effect.       ^    J^   2 

4    :    1  :  :  6    :     1       ^  ^  8 

8    :  []  ' 

3  1 16=5^  Am, 
Note,  Here  one  term  in  each  couplet  is  one  wall. 

Art.  33.  Similar  questions  sometimes  give 
rise  to  the  apparent  view  of  more  requiring  less. 
Money  must  be  compounded  with  time  before  it 
can  produce  interest,  as  more  money  will  require 
less  time  to  produce  the  same  interest ;  but  this 
is  all  apparent,  for  there  is  no  such  thing  as 
more  cause  producing  less  effect. 

Art.  34.  This  difficulty  arises  from  not  being 
able  to  readily  determine  which  is  cause,  and 
which  is  effect;  but  this  (the  only  difficulty  to  be 
met  with  in  this  system  of  statement),  is  readily 
overcome,  when  we  consider  that  all  action  of 
any  nature  must  be  cause,  and  that  which  is 
accomplished,  or  follows  such  action,  must  bo 
^ect. 


VADE   MECUM.  2* 

If  5  horses  in  8  days  consume  80  bushels  of 
oats,  how  many  bushels  will  3   -gQ  -j^q 
horses  consume  in  15  days  ?  ^i  g 

Cause.  Effect.   Cause.  Effect.  Jf^     3 

5    :    80  : :  3  :    []  , 

8   :  15  :  1  90  Am, 

Note.  Here  it  is  evident  that  the  action  of  the 
horses,  multiplied  by  the  days,  in  both  couplets, 
must  express  the  cause,  and  the  consumption  of 
the  oats  is  the  effect. 

Art.  35.  There  are  many  questions,  however^ 
where  it  is  indifferent  which  is  taken  for  cause, 
and  which  for  effect ;  only,  observe  when  one 
thing  is  taken  for  cause  in  the  first  couplet,  a 
similar  one  must  be  taken  for  cause  in  the  second 
couplet. 

If  the  transportation  of  4  cwt.  12  miles  cost 


810,  how  far  may  6  cwt.  be  carried 
for  815.  Ans.  12  M. 

Cause.  Effect.    Cause.  Effect. 

10    :    4  :  :  15  :  6. 
12::  []. 


X0 
0 


12 
12^. 


Or  thus 


Or  thus 


Cktuse.  Effect.     Cause.  Effect. 

4  :    10  :  :    6    :  15. 
12  :  []  : 

Cause.   Cause.     Effect.  Effect. 

4  :      6    : :  10  :  15. 
12:     []  :: 


9&  AHITHMETIOAL 

Or  thus : 

Effect.  Effect.     Cause.   Cause, 

15  :   10    :  :   6    :   4. 
[]  :  12. 

The  same  terms,  multiplied  together  in  each  of 
the  different  statements,  show  that  this  method 
16  strictly  scientific. 

1.  If  the  wages  of  6  men,  14  days,  be  $84, 
what  will  be  the  wages  of  9  men  for  16  days  ? 

Ans.  8144. 

2.  If  I  lend  8400  to  a  friend  for  16  months, 
how  long  ought  he  to  lend  me  81600  to  return 
the  favor  ?     (See  Art.  32.)  Ans.  4  mos. 

3.  If  2L  yds.  of  cloth,  2|  yds.  wide,  cost  83.35, 
how  many  yds.,  that  is  1^  yds.  wide,  can  I  have 
for  8134.00  ?  Ans.  160  yds. 

4.  If  11  men  in  7  days,  working  13  hours  per 
day,  dig  a  ditch  that  is  37|  ft.  long,  2|  ft.  wide, 
3^  ft.  deep,  in  how  many  days  can  5l  men, 
working  14  hours  per  day,  dig  another  that  is 
18|  ft.  long,  14f  ft.  wide,  and  10^  ft.  deep? 

Ans.  120 J  days. 

Kote.  In  stating  all  questions  in  Cause  and 
Effect,  see  Art.  25. 

Practice. 
The  preceding  principles  of  Cause  and  Effect 
will  apply  to  the  solution  of  all  questions  arising 
in  Practice,  which  is  nothing  more  than  Simple 
Proportion,  having  1  for  the  first  term. 

Interest. 
We  omit   the   definitions  of  terms   ■'^^^^   *- 


VADE    MECUM.  2J 

treating  of  Interest,  they  being  so  common  that 
the  learner  is  supposed  to  be  familiar  with  them. 

Art.  36.  The  system  of  Cause  and  Effect  is 
very  extensive  and  easy  in  its  application.  It 
covers  every  case  that  can  arise  under  Interest. 
To  find  the  interest,  principal,  time,  or  rate  per 
cent.,  and  thus  dispense  with  five  or  six  special 
rules,  as  found  in  almost  every  Arithmetic,  w« 
will  use  the  following 

Interest   Table. 

CauHK.  Effect.  Cause.  Effect. 

100     :  Ptate  pr.  ct.   :  :  Principal  :  Interest, 
lyear:*  Time  : 

.^^"  Make  the  blank  lohere  the  table  designates 
(he  term,  or  factor,  you  wish  tojind. 

"What  is  the  interest  of  8200,  for  3  years,  at 
6  per  cent.?  ^2^ 

Cause.         Effect.  Cause.         Eff'Ct.  ^VPi    3 

100     :      6     :  :    200     :     "[]  ^    6 

1       :  3       :  I 

IS36  Ans. 

Art.  37.  What  principal  at  interest  for  24 
months,    at    6    per    cent.,   will  ,  -^    o 

gain84S?  ^'^^    ^ 

Cause.         Effect.  Cause.         Effect.       r^\ 

100     :      6      ::     f]      :     48  — 

12      :  24      :  |8400   An*. 

*  Or  the  factors  of  a  year.  If  the  time  be  months,  12  ;  if 
days,  12  and  30 ;  if  weeke,  52.  etc 


llOO 
12 


ARITHMETICAL 


Art.  38.  At  what  rate  per  cent,  will  $200,  in 
450  days,  gain  815  interest  ? 

Cause     Eject.  jj^0 


Cause.     Eject, 

100 
12 
30 


[] 


200 
450 


15 


X$    6 


6  pr.  ct.  An$. 


Art.  39.    In  wliat  time  will  $160  gain  $2 


interest  at  3  per  cent  ? 


Cause.    Effect.       Cause.    J 

Wect. 

100   :     3     ::   160  : 

2 

]2    :                 []    : 

J0P  5 


5  mos.Ans. 

Note.  Many  abridged  rules  miglit  be  given 
for  tbe  solution  of  interest  questions  ;  we  shall, 
however,  give  but  few,  as  we  are  satisfied  that 
those  who  make  themselves  acquainted  with  the 
preceding  general  principles,  will  be  able  to  make 
their  own  abridgments. 

Art.  40.  1.  When  the  time  is  months,  and 
rate  per  cent.  6,  to  find  the  interest,  multiply  the 
principal  by  half  the  number  of  months. 

2.  When  days,  divide  them  by  60,  and  multi- 
ply the  quotient  by  the  principal. 

3.  When  the  time  is  months,  and  the  rate  per 
cent.  4,  multiply  the  principal  by  ^  the  number 
of  months. 

4.  When  the  time  is  months,  and  the  rate  per 
cent.  3,  multiply  the  principal  by  \  the  number 
df  months. 


VADE    MECUM.  29 

5.  To  find  tlie  interest  of  any  jrincipal,  for 
any  time,  at  any  rate  per  cent : 

Make  a  dividend  of  the  principal,  time, 
and  rate  per  cent.  If  the  time  he  months,  the 
divisor  is  12  ;  if  days,  12  and  30,  etc. 

Note.  When  the  time  is  given  in  different 
denominations  —  as  months,  days,  etc.  —  it  must 
first  be  reduced  to  the  lowest  denomination  men- 
tioned, then  placed  on  the  line. 

Miscellaneous  Examples. 

1.  What  is  the  interest  of  $100,  for  3  years, 
at  5  per  cent.  ?  Ans.  815.00. 

2.  What  is  the  interest  of  821,  for  1  year  and 
4  months,  at  3  per  cent.  ?  Ans.  84  cts. 

3.  What  is  the  interest  of  850,  for  18  months, 
at  4  per  cent.  ?  Ans.  83.00. 

4.  What  is  the  interest  of  884,  for  6  months 
and  20  days,  at  9  per  cent.  ?  Ans.  84.20. 

5.'  What  is  the  interest  of  875,  for  60  days, 
at  8  per  cent.  ?  Ans.  81.00. 

6.  What  is  the  interest  of  846,  for  90  days, 
at  6  per  cent.  ?  Ans.  69  cts. 

Note.  To  save  space  in  giving  other  problems, 
use  the  interest  in  the  preceding,  and  find  the 
priiLcipal,  time,  and  rate  per  cent.,  alternately,  by 
the  Interest  Table  (Art.  36). 

Percentage. 

Art.  41.  To  find  the  amount  for  which  an 
article  must  be  sold,  to  gain  or  lose  any  given 
rate  per  cent. 


30  ARITHMETICAL 

State  thus:  As  100  is  to  100  with  the  gain  per  cent, 
added,  or  loss  per  cent,  subtracted,  so  is  the  prime 
cost  to  the  required  price. 

1.  A  merchant  paid  44  cents  per  yard  for  cloth, 
for  what  must  he  sell  it  to  gain  25  per  cent.  ? 

Cause.         Effect.  Cause.       Effect.    ^  ^W\  ^^    \\ 

100  :  1004-25::    44    :     []  

1^.  55  cts. 

2.  Paid  ^80  for  a  horse,  for  what  must  he  be 
sold  to  gain  50  per  cent.  ?  Ans.  ^120. 

3.  Paid  $90  for  a  horse,  how  must  he  be  sold 
to  lose  33 i-  per  cent.  ? 

?  i^p    30 

Caxish.  Effect.  Cause.      Effect.      I0pl200 

100    :  100—33^  ::    90     :    [].  | 

I  $60  Ans, 

4.  Paid  6100  for  sheep,  how  must  they  be 
sold  to  lose  20  per  cent.  ?  Ans.  880. 

Art.  42.  Having  the  cost  and  selling  price 
given  to  find  the  gain  or  loss  per  cent. 

State  thus  :  100  is  to  the  required  gain  or  loss 
per  cent,  as  the  prime  cost  to  the  difference  between 
the  prime  cost  and  selling  price. 

1 .  A  merchant  bought  cloth  at  48  cts.  per  yd., 
and  sold  it  at  60  cts.  what  was 
the  gain  per  cent. 


^ 


Cause. 

Effect. 

Caute. 

Effect. 

100 

[] 

::     48 

:  60—48 

JP0  25 


.4.25  cts. 


VADE    MEOUM.  SI 

2.  Had  he  paid  60  cts.,  and  gold  it  for  48  eta., 
what  would  have  been  the  loss 

per  cent.  ?  ^^1  X$  2 

Gaute.      Ejfect.        Caute.  Effect.  \ 

100     :     []     :;     60     :  60—48  U.20ct8. 

3.  A  farmer  paid  875  for  a  horse,  and  8150 
for  a  chaise;  he  sold  the  horse  for  8100,  and  the 
chaise  for  8125.  What  per  cent,  did  he  gain  on 
the  horse,  and  Jose  on  the  chaise  ? 

A         \^^}^  per  cent,  gained. 
(  16|i?cr  cent,  lost, 

Art.  43.  Having  the  selling  price  of  an  article, 
and  the  rate  per  cent.,  gained  or  lost,  given,  to 
find  the  cost. 

State  thus  :  100  t*  to  100  with  the  gain  per  cent, 
added,  or  loss  per  cent,  subtracted,  as  the  required 
cost  to  the  the  selling  price. 

1.  Having  sold  a  watch  for  814,  I  thereby  lost 
30  per  cent.,  what  did  it  cost  me  ? 

Oaute,  Effect.  Cause.     Effect.         '^"jlO0 

100  :    100—30     ;:    []    :   14. 


120  A 


ns. 


2.  If  a  farm  be  sold  for  8220,  and  10  per  ct. 
is  gained,  what  did  it  cost  ? 


100 

m 

100   :    100+10  ::     []      220.        &  Ans. 


Caute.  Effect.  Caute     Effect.         V^^ 


S2 


ARITHMETICAL 


3.  I  sold  a  horse  for  $40,  and  by  so  doing 
lost  20  per  cent. ;  whereas,  I  ought,  in  trading, 
to  have  cleared  30  per  cent.  How  much  was  he 
sold  ior  unuer  iiis  real  value  ? 

V)0~20:      100    ::  40  :  n=50 


:[]: 


lO'J     :100-f30::  50  :  []=65— 40=.^25^»# 

Or  thus  :       J:pp,X?0  65 
I  ^j3 


65—40=825  Am. 


Barter, 

Art.  44.  (See  Article  32).  The  term  under- 
stood in  Barter  is  one  amount.  *^ 

1.  How  many  bushels  of  apples  must  I  have, 
at  25  cts.  pr.  bu.,  for  35  yds.  of  calico,  at  20  cts. 
per  yard  ?  ^  .  ^ 

Cause.        Effect,         Cause,        Effect.     ^   ?^  ?^   J 

1      ; :     35     :      1  W^ 


II  ; 


20  I  28  In.. 


2.  If  I  barter  12  bu.  of  flaxseed,  at  65  cts.  pr. 
bu.,  for  cloth,  at  60  cts.  pr.  yd.  How  many 
yards  must  I  have  ? 
Cause.        Effect.        Cause.        Effect.     P  W 

12      :       1      ::     []       ::      1 

65     •  60     ::  13  ^n# 


^^  13 


VADE    MECUM.  33 

DiscouTit. 

Art.  45.  State  thus:  As  100  is  to  the  amount  of 
100 /or  the  given  time,  at  the  given  rate  per  cent., 
90  is  the  required  present  worth,  to  the  given  debt. 

Xote.  Subtracting  the  present  worth  from  the 
amount,  will  give  the  discount, 

1,  What  is  the  discount  of  $436,  for  18 
months,  at  6  per  cent.  ? 

Cause.     Effect.     Cause.     Effect.  ^^^   ^ 

100    :  109  ::  []    :  436.  r; ^  ^^^^ 

'-■'  \Pres.  worth  8400 

436—400=36  Ans. 

The  following   process   is   preferable    for  its 

brevity ; 

Make  a  dividend  of  the  2^rincipal  and 

time;    multiply    one  year    (or   its   equivalent  in 

months  or  days)  by  100  ;    divide  by  the  rate  per 

cent.,  and  add  the  time  (of  the  same  denomination) 

to  the  quotient,  for  a  divisor, 

N.  B.  The  result  will  be  the  discount  in  the 
lame  denomination  of  the  sum  given. 

To  find  the  present  worth,  subtract  the  di§- 
count  from  the  sum  given. 
Take  the  preceding  example  : 

12xlOO-^6=200.     200+18=218,  divisor. 

PPM^  2 
18 


I  $36  discount, 
3 


84'  ARITHMETICAL 

To  prove  discount. 

The  interest  of  the  present  worth  must 
equal  the  discount  of  the  sum,  for  the  same  tm^. 
at-  the  same  rate  per  cent. 


836  interest=^'^Q  discount. 

2.  What  is  the  discount  of  ^80,  for  200  days, 
at  12  per  cent.  ? 
360x100-^12=3000.    3000+200=3200  c/^V'r. 


X^  ?^0P 


80    5 


$5  Ans. 


3.  What  is  the  discount  of  $48,  for  4  years, 
at  5  per  cent.?  Ans.  $8. 

4.  What  is  the  discount  of  8218,  for  9  mos., 
at  12  per  cent.  ?  Ans.  818. 

5.  What  is  the  discount  of  81140,  for  120 
days,  at  4  per  cent.  ?  Ans.  815. 

6.  What  is  the  difference  between  the  interest 
of  8160,  for  400  days,  at  6  per  cent,  and  the 
discount  of  the  same  sum,  for  the  same  time,  at 
the  same  per  cent.?  Ans.  66|  cts. 

Measurement. 

Art.  46.  For  all  measurement,  use  the  follow- 
ing table  in  stating  —  or  rule,  if  Cause  and  EffetU 
Ise  read  alternately : 


VADE 

MECUM. 

59r 

Mensuration  Table. 

Cam  St. 

KjftcL. 

Cause. 

Eject, 

Fioion!  Of  tue 

unit  of 

measure. 

Unit 

of 

measure. 

Factors  of  the 
::         thing   to 
be  measured. 

Number 
of 

units.* 

S^  Make  the  blank  in  the  term  where  the  tablt 
designates  the  term  or  factor  sought. 

Cord   Wood. 

Art.  47.  1.  Ilowmany  cords  are  there  in  a 
pile  of  wood  80  feet  long,  4  feet  wide,  ,8  feet 
deep  ? 

8  $0  20 


luse. 

Eject. 

Cause. 

Eject 

8 

:      1      : 

:    80 

[]• 

4 

4 

4 

8 

I  20  Ans. 

2.  How  mncli  will  a  pile  of  wood  cost,  that 
is  48  ft.  long,  2\  ft.  wide,  and  16  ft.  deep,  at  $1 
per  cord  ?  Ans.  §15.00 

3.  How  long  must  a  pile  of  wood  be  to  con- 
tain 13  cords,  that  is  6^  ft.  wide,  and  4  ft.  deep  ? 

X^\  2 


Cause. 

8 
4 
4 


Eject. 
1 


Cause. 

R 

4 


Eject. 

13. 


8 
4 
164  Ans. 


*The  number  of  units  is  the  thing,  or  answer,  sought ;  but 
If  this  be  givcB  some  factor  is  sought. 


86 


ARITHMtllCATi 


4.  How  wide  must  a  pile  of  wood  be,  that  ia 
24  ft.  long,  and  16  ft.  deep,  to  contain  12  cords  ? 

Jins.  4  ft. 

Land  Measure, 


Art.  48.  1.  How  many  acres  of  land  in  a  field 
80  rods  square?  ^  ^^  20 

Cause.        Eject.  Cause.        Effect.        ^0  g0   2 

4      :      1      ::      80      :     [].  . 

40     :  80      :  A^  Ans. 

2.  What  is  the  area  of  a  rectangular  field  60 
rods  long,  and  121  yards  wide  ? 


Cause. 

4 

40 
5^ 


Effect.         Cause.        Effect.    ^  ^p 

1      :  :     60    :      [].        XX 
121   : 

4 


^0 

3 

xp 

11 

^ 

33 

Ans.  S\, 

3.  How  wide  must  a  rectangular  lot  be  that  ie 
24  rods  long,  to  contain  3  acres  ? 

3  $fi 


Cause. 

Effect. 

Cause. 

Effect 

4 

:      1     : 

:     24 

:      3. 

40 

: 

[] 

: 

20  Ans. 


Nate.  This  principle  is  applicable  to  the  meas- 
urement of  lands  of  all  shapes,  as  given  under 
Mensuration. 


VADE    MECCM. 


Carpeting. 

Art.  49.    1.  How  many  yards   of  carpeting, 
t^at  is   ^    of  a  yard   wide,  will  be  required  to 


cover  a 

floor, 

27  feet  long, 

and  13  feet  wide? 

^P 

Cause. 

I^ect 

Cause. 

Effect. 

^|13 

1      : 

1 

:  :    27      : 

[J- 

4 

5 

13      : 

— 

9  152  yds.  ^. 

Xote.  The  9  under  theT^r*-^  cause  is  to  reduce 
it  to  square  feet,  to  be  of  the  same  denomination 
of  the  second  cause. 

2.  How  wide  must  a  floor  be,  that  is  18  feet 
long,  to  require  12  square  yards  to  cover  it  ? 


Cause. 

Effect. 

Cause. 

Effect. 

1 

:      1      : 

:    18 

:     12 

1 

[] 

: 

9 

X^ 


\Xf  6 


6ft.^. 


8.  What  will  it  cost  to  carpet  a  room  that  is 
36  feet  long,  10  feet  wide  —  carpeting  1^  yards 
wide,  and  worth  37^  cents  per  yard  ? 


Cause. 

E^ect. 

Cause. 

Effect. 

# 

j'p 

1 

:      1      : 

:  36 

:  tJ- 

^ 

75 

1\ 

: 

10 

4 

9 

37i 

§12.00  A. 

Crib  and  Box  Measure. 

Art.  50.  1.  How  many  barrels  of  corn  will 
a  crib  bold,  that  is  50  ft.  long,  9  ft.  wide,  and  3 
ft.  4  in.  deep  ? 


B8 


i 

<k 

ARITHMETICAL 

^;^  2 

itise. 

Elect. 

Came.        Effect, 

?> 

•; 

^k 

:     1     : 

:50           :     []. 

V5 

^    3 

5 

9 
3a4in  : 

10 
]20bbls.  .1. 

2.  How  many  bushels  of  corn  will  a  crib  hold, 
that  is  18  ft.  9  in.  long,  8  ft.  high,  7  ft.  6  in.  wide  ? 


Cause. 

2k 


Effect. 
1 


Cause. 

8 
18ft9in 
7ft6in 


Effect. 
[] 


^12 


150  bu.  A. 


3.  How  high  must  a  crib  be,  to  contain  dGO 
bushels  of  corn,  when  it  is  18  feet  long,  and  6 
feet  3  inches  wide  ? 

Cause.         Effect.  Cause.         Effect.  ^^K^^       ^  . 

2k     :      1      ::     18      :    860       .mWn^ 
6ft3in 
[]      : 


^  m  2 


I8ft.^«« 


4.  How  many  bushels  of  wheat  in  a  box,  10 
feet  long,  3  feet  wile,  and  2  feet  8  inches  high  ? 


Cau$e. 


Effect. 
1 


Cause. 

10      : 
3      : 

2ft8in 


Effect. 


^0  2 

jr^i32 


l64  ^«#. 


VAJJE    ME  CUM. 


d9 


5.  How  lonsr  must  a  box  be,  that  is  4  feet  2 


inches  wide,  and 
50  bushels  ? 


feet  1  inch  dee}),  to  contain 

4|5 


Cause. 


Effect. 
i 


Canae.  Effect. 

4  ft.  2  in.  :  50 
2  ft.  1  in.  : 

[] 


^12 


on 


36 


I'jft.A. 


6.  How  many  panes  of  glass  in  a  box  of  50 
feet,  8  inches  by  10  inches  ? 


Cause. 


10 


ffect. 

Cause. 

Effect. 

1      : 

:      50 

1  2 

I  2 

''  n 

ip 


X^  3 
1^  6 
5p 


90^. 

7.  How  many  feet  of  glass  in  a  box,  that  con- 
tains 120  panes,  10  in.  wide,  and  12  in.  long? 

^^  10 


Cause. 

Effect. 

Cause. 

Effect. 

10 

:       1 

■■■■    [] 

:  120. 

12 

1  2 
1  2 

X^ 


jr^ 


100  ^n5. 

y  yote.  The  12s  under  the  second  cause  in  the 
preceding  examples  are  to  reduce  them  to  square 
inches,  to  be  of  the  same  denomination  of  the 
Jirsi  cause. 

House  Covering. 

Art.  51.  1.  How  many  shingles  will  be  re- 
quired tD  cover  a  house,  that  is  24  ft.  long,  and 
15  ft.  wide? 


40 

ARITHMETCAL. 

^ 

12 

Cau»e. 

4     : 
6     : 

Effect. 
1 

Oavse.         Effect. 

J 

12 

^ 
? 

12 

n  5 

4 

2880  An^ 

Note*  It  is  customary  to  allow  the  shingles 
to  be  4  inches  wide,  and  to  show  6  inches,  and 
the  rafters  to  be  |  the  width  of  the  house,  both 
making  \.  When  this  is  the  case,  the  process 
may  be  shortened  by 

Multiplying  the  product  of  the  length  and  width 
of  the  house,  by  8. 

2.  How  many  shingles  will  be  required  to 
cover  a  building,  that  is  30  feet  long,  and  25  feet 
wide  ?  Ans.  6000. 


Art.  52.  1.  How  many  boards  will  be  re- 
quired to  cover  a  house  36  ft.  long,  24  ft.  wide ; 
tiie  boards  6  in.  wide,  and  to  show  18  inches? 


'ause. 

6 

18 

Effect. 

:      1      :  : 

> 

Cause. 

36     : 
24     : 

Effect.     ^$ 
[] 

Xf  2 
X^  4 
24 

4 

1536  Ans 

2.  How  much  must  I  pay  for  boards  6  inches 
wide,  and  to  show  15  inches,  at  86  per  1000,  to 
cover  a  house  24  feet  long,  and  20  feet  wide  ? 


YADA    MEC 
Cause.  Effect.    Cause.  Effect, 

6  :  1  :  :  24    :    []. 
15:            20    : 

lOOO  :  6  : :   1     : 

12 

ja 

'CM. 

5    X^ 

25JPPP 

X%  4 
X%  4 
24 
2p 
4 

125 

768 

41 


86  14c.  4m 


Brklc    Work. 


Art.  53.  1.  How  many  bricks,  8  inclies  long, 
4  inches  wide,  will  be  required  to  pave  a  walk 
3  feet  4  inclies  wide,  and  ^  of  a  mile  long  ? 


^\X^ 

k  \x^ 

X$\4p  10 

1 

3 

11 
40 


Caiuc, 

Effect. 

Cause. 

Effect, 

8        : 

1     : 

'       \ 

:      []• 

4 

3ft4in 

12 

'I 

4  0 

8 

13200^. 


2.  How  many  bricks  will  be  required  to  build 
tbe  walls  of  a  house  20  feet  long,  15  feet  wide, 
16  feet  high,  and  8  inches  thick,  allowing  \  for 
mortar,  the  brick  to  be  8  inches  long,  4  inches 
wide,  and  2^  inches  thick  ? 


42 


ARITHMETICAL 


Cause. 

Effecx. 

Cause. 

8 

.       1 

::     35 

4 

: 

2 

2| 

16 

8 

5 
6 

I  2 

12 

Effect. 
[]• 


$  12 

85 

2 

16 


13440^. 

^Vb^c.  Adding  tlie  length  and  width  together, 
and  doubling  the  sum,  gives  one  straight  wall. 
Multiply  that  by  the  height  and  thickness  and  by  f, 
(making  a  deduction  of  \  for  mortar). 

N.  B.  Deductions  must  be  made  for  windows 
and  doors. 


Log  and  Plank  Measure. 

Art.  54.  1.  How  many  solid  feet  in  a  log,  21 
inches  in  diameter,  and  16  feet  lorn 


Cans 

12 
12 

12 


Effect.        Cause.         Effect. 

1      ::    21      :      []. 
21 

16 

1  2 


?  x^ 
px^ 

2  X^ 

X$ 
^X7 

11 

x^  ^ 

2 

77 

38L^ 

Note.  To  find  the  solid  contents  of  a  log,  we 
first  find  the  area  of  the  end  by  Art.  68,  and 
multiply  by  the  length. 


VADE    MECUM. 


43 


2.  How  many  square  feet  of  plank,  for  ceiling, 
flooring,  etc.  (1  inch  thick),  in  a  log,  24  inches 
in  diameter,  20  feet  long,  allowing  \  for  saw- 
calf? 


7ause. 

Eject. 

Cause. 

12      : 

t— 1 

24 

12      : 

24 

H 

20 

2 

1  a 

Xote. 

Dividin 

s:by2 

Effect. 


^124 
?0  4 


384  ^n.9. 

ws  away 
in  getting  the  area  of  the  end.     (See  Art.  61  ) 

3.  How  many  square  feet  of  plank,  1  inch 
ihick,  in  a  log  30  inches  in  diameter  and  12  feet 
long,  allowing  ^  for  saw-cut  ?  Ans.  360. 

4.  How  many  square  feet  of  plank.  1  inch 
thick,  in  a  log  48  inches  in  diameter,  and  10  feet 
long,  allowing  ^  for  the  saw-cut  ?         Ans.  768. 

5.  How  many  square  feet  of  sheeting  plank,  \ 
of  an  inch  thick  (including  saw-cut),  in  a  log, 
12  feet  long,  and  7  inches  in  diameter  ? 


Cause. 

12 

1  2 
3 

4 


Effect. 
1 


Cause 

12 

7 
7 

vv 


Effect. 
[] 


3U 

?  xm^ 

Xfl 
11 


2 


X^  _ 
154 


51i-  Ans. 


6.  To  cut  7|-  square  feet  off  of  a  plank  6  inches 
wide,  how  many  feet  of  it€  length  must  be  taken  ? 


u 


Cause, 

12 
I. 


ARITHMETICAL 


EJecL 
1 


Cause, 

I 


Eject, 
1\, 


m 

^15 


!l5  Ans. 


Promiscuous  Examples. 

1.  How  many  acres  are  tliere  in  a  round  field, 
56  rods  in  diameter  ? 


Cause, 

4 
40 


Effect. 
1 


Cause. 

r;6 

5(3 


Effect. 


4  m  7 
xm 


77 


15|  A, 


2.  If  I  send  12  bushels  of  wheat  to  mill,  how 
many  pounds  of  Hour  will  I  get,  allowing  ^^  for 
toll,  \  for  bran,  \  for  shorts  ;  weight  of  wheat, 
60  lbs.  per  bushel  ? 


Cause, 

12 

8 
6 


Effect. 

Cause. 

:      11     : 

:      12     : 

:        7     : 

:      60     : 

:        5     : 

Effect. 


xi 

11 

4    ? 

7 

0 

5 

Jr? 

^j3^p5 

4 

1925 

48H^ 

8.  I  wish  to  get  481i-  lbs.  of  flour  ;  how  many 
bushels  of  wheat  must  I  send  to  mill,  making  tlie 
eame  allowances  as  in  the  preceding  example  ? 


VADE    3IECCM. 


45 


Chuse. 

12 

8 
6 


Effect. 
11 

7 
5 


Cause. 

y 


481V 


12  ^>w. 


4.  How  many  barrels  of  corn  in  a  field  240 
hills  long,  by  160  wide,  each  hill  to  average  2 
ears,  120  of  which  will  make  1  bushel  ? 


Cause. 

Effect. 

Cause. 

120 

:      1     : 

:     IGO 

5 

'           ' 

:     240 

2 

Effect. 
[]• 


m 


^^p     2 
jr0P  32 
2 


^.  128 


MENSUEATIOX  OP  SUPERFICES. 


Art.  55.  To  measure  a  square.  ^ 


Hule.     Multiply  the  side  A  B 
irdo  A  D. 

Let  AB=12.      AD=12. 
Then,  12X12=144.  D 


48  ARITHMETICAL 

Note.     The  area  will  be  of  the  same  denomi- 
nation that  the  sides  are. 

Art.  56.    To  measure  A B 

a  rectangle. 

Rule.  Multiply  the  length 
hy  the  width.  D 

Let  A  B=40.     A  D=12. 
Then,  40x12=480. 

Art.  57.  When  the   area  and   one   side  are 
given,  to  find  the  other. 

Rule.     Divide  the  area   [reduced  to  the  same 
denomination  as  the  side),  hy  the  given  side. 

In  the  last  fi^ire,  A  B=40,  and   area=480, 
to  find  A  D.     480-^0=12. 

Art.  58.    To  measure  a  ^ B 

rhombus.  xT  |\ 

\  '  \ 

Rzde.     Multi'ply  one  side         \            :     \ 
hy  the    shortest   distance  be-            \           l       \ 
tween  the  sides.  V U ^ 

LetAB=16.     BE=12. 
Then,  16x12=192. 


Art.  59.  To  meas- 
ure a  rhomboid. 


Rule.     Multiply  one  D       E  C 

of  the  longer  sides  by  the  shortest  distance  between 
them. 


TADE    JrECUM. 


47 


LetAB=40.    A  E=16. 
Then,  40x10=040. 


Art.  60.  To  measure 
a  trapezoid. 

Eule.  Multiijhj  the 
half  sum  of  the  parallel 
sides  by  the  shortest  dis- 
tance between  them.  iT 


LetAB  =  8       DC  =  18.     B  E=10. 
Then,  18+8-MixlO=130. 


Art.  61.    Tlie  diagonal*  of  a 
square  given  to  find  the  area. 

Ride.     Multiply  the   diagonal 
hy  half  itself. 

Let  B  D=SO. 
Then,  80x40=3200. 


Art.  62.  To  measure  a  right- 
angled  triangle. 

Ride.     Multiply   the   base   by 
ik€  perp)endicidar ,  and  tale  half. 

LetAC=16.     BC=19. 
19x16h-2=152. 


48 


ARITHMETICAL 


Art.  68.  To  measure  an  acute 
or  obtuse-angled  triangle. 

Rule.     Multiply  the  base  hy  a 
perpendicular  line  from  the  vertex  ^ 
to  the  base,  and  take  half. 


LetAC=60.     BD==24.    60x24-^2=720. 
I^ote.     Take  the  longest  side  for  the  base. 


Art.  64.  To  measure  any  regular 
polygon. 

Rule.  Multiply  one  side  by  the 
perpendicular  distance  from  the 
center  ;  take  half  the  product,  and 
multiply  the  quotient  by  the  number 
of  sides. 

Let  AB=15,  ab=20,  (No.  of  side 
15x20-^2x8=1200. 


S). 


Art.  65.  To  measure 
a  trapezium. 

Ride.    Draio  a  diag- 
onal line,  and  calculate 

the  two  triangles  by  Art.  C 

63,  t/ieir  sum  will  be  the  area  of  the  trapezium. 


VADE    ME CUM. 


49 


Art.  66.  To  measure  any  irregU' 
lar  figure. 

Rule.     First  cut  it  into  triangles 
hy  drawing  diagonal  lines.      Calcu- 
late   [by    Article    63)    the   several 
triangles,  and  tJieir  sum  will  he  the  area  of  tht 
figure. 

Art.  67.  To  measure  a  circle. 
Given,  the  diameter  of  a  circle, 
to  find  the  circumference.  A 

Rule,  Multiply  the  diameter 
byS'. 

Let  AB=14.  14  XV  =44. 

A^ote.  When  the  circumference  is  given,  to 
find  the  diameter,  multiply  the  circumference  by 
/o,  (the  converse  of  the  preceding.) 

Let  A  a  B  b=44,  to  find  A  B.     44x/2=14. 

Art.  68.  To  find  the  area  of  a  circle. 

Rule.  Multiply  the  circumference  by  the  dianu- 
ter,  and  take  one-fourth. 

In  the  preceding  44x14  .'  1=154, 
Or,  multiply  the  square  of  the  diameter  by  \\. 

Art.  69.  To  find  the  diameter  of  a  circle  equal 
to  a  square  whose  side  is  given. 

Rule.     Multiply  the  side  by  1.128. 

Let  the  side=10.     Then,  10x1.128=11.280. 


50 


ARITHMETICAL 


Art.   70.    To   find  tlie 
area  of  an  ellipse. 

Bule.    Multiply  the  pro-  a 
duct  of  the  transverse  and 
conjugate    diameters   (A  B 
and  a  b)  by  \\. 

Let  A  B=14.     a  b=10. 

Art.  71.  To  find  the  area  of  a 
globe. 

Rule.  Multiply  the  circumfer- 
ence by  the  diameter ;  or,  multiply 
the  square  of  the  diameter  by  \- . 

Art.  72.  To  find   the  area  of  a  cylinder,  or 

round  body  of  equal  largeness  from  end  to  end. 

Rule.     Multiply  the  circumference  by  the  length. 

Art.  73.  To  find  the  area  of  a  right  cone. 
Rx.le.     Multiply  the  circumference  of  the  base 
by  the  slant  height,  and  take  half. 


MENSURATION   OF   SOLIDS. 


Art.  74.    To  find  the  solidity 
of  a  right-angular  solid. 

Rule.      Multijyly    the    length, 
breadth,  and  depth  together. 


VADE    MECUM. 


51 


Let  tlie  lengtli=20,   widtli=12,  ]ieiglit=10. 
20x12x10=2400. 

Art.  75.    To  find  the  solidity  of  a  cylinder  or 
prism. 

Rule.     Multiply  the  area  of  the  base  (or  end), 
by  the  length. 

Xote.     Find  the  area  of  the  base  by  previous 
rules,  according  to  its  shape. 

Art.  76.  To  find  the  solidity  of  a  solid  wedge. 

Rule.     Multiply  the  area  of  the  base  by  half  the 
perpendicular  length. 


Art.  77.  To  find  the  solidity  of  a  "pyramid  or 
cone. 

d 

Rule.     Miiltiply  the  area  of  the 
hase  by  one-third  of  the  altitude. 

Let  a  b=14,  c  d=24. 


m  2 


4! 


22 
14 

^4  2 


11232^^5. 


Art.  78.   To  find  the  solidity  of  the  frustrum 
of  a  pyramid,  or  cone. 


52  ARITHMETICAL 

Rule.  Add  the  areas  of  the  upper,  the  lowers 
and  the  middle  bases  tor/ether,  (the  middle  base  it 
found  by  multiplying  the  upper  andlower  bases 
together,  and  extracting  the  square  root  of  the  pro- 
duct,) and  multiply  the  sum  by  one-third  of  the 
altitude^ 

Art.  79.  To  find  the  solidity  of  a  globe. 

Rule.  Multiply  the  area  by  one-sixth  of  the 
diameter ;  or,  multiply  the  cube  of  the  diameter 
by  \\. 

Art.  80.  To  find  the  solidity  of  a  spherical 
segment. 

d 

Rule.     Add  the  square 
of  the  height  to  three  times 
the  square  of  the  semi- 
diameter    of    the    base, 
and  midtiply  the  sum  by 
the  height,   and  by  ll. 

Let  a  b=10,  c  d=4. 

5X5X3+4X4=91.     91x4XH=190H^»*. 


SQUARE  ROOT. 

Art.  81. — Rule.  1.  Separate  the  given  nmnber 
into  periods  of  two  f  mires  each,  commencing  with 
Knits. 


VADE    MECDM.  53 

2.  Find  the  greatest  root  in  the  left  hand  period, 
and  place  it  on  the  right.  Subtract  its  square 
from  the  Jirst  period,  and  to  the  remainder  bring 
down  the  next  period ;  and  make  a  dividend  of 
the  remainder ,  with  the  first  figure  of  the  period 
annexed. 

3.  Double  the  root  for  a  divisor,  and  set  the 
quotient  in  the  root,  and  to  the  right  of  the  divisor. 

4.  Multiply  and  subtract  as  in  division,  and 
proceed,  as  before,  until  all  the  periods  are  brought 
down,  when  periods  of  noughts  may  be  annexed  to 
obtain  decimals. 

1.  What  is  the  square  root  of  55225  ? 
5'52'25  (235=roo/ 


43)1  52 
129 


^^^^    23  25 

2.  What  is  the  square  root  of  15625  ?    ^.125. 

3.  What  is  the  square  root  of  5'35.92'25  ? 

Ans.  23.15. 


CUBE  ROOT. 

Art.  82. — Rule.  1.  Separate  the  given  nurriber 
into  periods  of  three  figures  each,  commencing  with 
units. 


54 


ARITHMETICAL 


2.  Find  the  greatest  root  in  the  left  hand  period^ 
2^lace  it  on  the  right,  and  subtract  its  cube  from  the 
left  hand  period. 

3.  To  the  remainder  bring  down  the  next  period^ 
and  make  a  dividend  of  the  remainder  and  the  first 
figure  of  the  period  annexed. 

4.  Multiply  the  square  of  the  root  by  3,  for  a 
iefedive  divisor.  Place  the  quotient  in  the  root, 
and  its  square  to  the  right  of  said  divisor,  supply- 
ing the  pilace  of  tens  with  a  cypher,  if  the  square 
be  less  than  ten. 

5.  Complete  the  divisor  by  adding  to  it  the  pro- 
duct of  the  last  figure  in  the  root  by  the  rest,  and 
iy  30. 

6.  Multiply  and  subtract  as  in  division,  and 
bring  doivn  the  next  period. 

7.  Find  the  next  defective  divisor,  by  adding  to 
the  last  complete  divisor  the  number  which  com- 
pleted it,  ivith  twice  the  square  of  the  last  figure  in 
the  root,  and  proceed  until  all  the  periods  are 
brought  down,  when  decimals  may  be  found  by 
annexing  periods  of  noughts. 

1.  What  is  the  cube  root  of  9663597  ? 
9'663'597(213 


2x2 X  3=1201 
1X2X30=     00 

Complete  div 1261 

IXlX  2=       2 

Defect.  rfiyV....132309 
21X3X30=1890 

Complete  div. ..l^^^l^ 


1663 


\2^\=subtrahend. 


U2597 


A02hTi=subtrah€nd, 


VADE    MECUM. 


03 


2.  What  is  the  cube  root  of  1953125  ? 

Ans.  125. 

3.  What  is  the  cube  root  of  33131834.347032  ? 

Ans.  321.18. 

We  shall  facilitate  the  rules  of  square  and  cube 
root  by  the  following  properties  : 

1.  The   product  of  two  square  numbers  is  a 
square  number. 

2.  The  quotient  of  two  square  numbers  is  a 
square. 

3.  The  product  of  two  cube  numbers  is  a  cube. 

4.  The  quotient  of  two  cube  numbers  is  a  cube. 


Table  of  Sqn 

ares  and  Cubes. 

Nos. 

Squ'rs. 

Cubos. 

1|2    3|  4|     5 
1|4    9  161  25 
l|8!27|64il25 

6       7|      8 

36    49|    64 

216|843  512 

9      10 

81    100 

729!  1000 

(See  Art.  30.)  In  order  to  work  with  speed 
and  alacrity,  attention  and  tact  are  necessaiy  on 
the  part  of  the  learner. 

1.  What  is  the  side  of  a  square  piece  of  land 
containing  360  acres  ? 

Instead  of  multiplying  by  160,  and  extracting 
the  square  root,  according  to  the  common  method, 
Ave  lemove  a  nought  from  one  to  the  other, 
making  them  both  squares,  whose  roots  are  60 
and  4,  or  6  and  40. 

Maltiply  the  roots  together.    60  X  4=240  Ana. 


§6  ARITHMETICAL 

2.  What  is  tlie  square  root  of  the  product  of 
32  and  128  ? 


i'^ 


n  2 


716=4x16=64^725. 


Here  it  will  be  observed  that  both  numbers 
are  divided  by  the  factor  16,  and  the  root  of  the 
product  of  the  quotients  multij)lied  by  the  factor. 

N^ote.  We  should  examine  the  table  closely, 
so  as  to  recognize  a  square  or  cube  as  soon  as 
seen.  The  principle  of  removing  noughts,  or 
using  factors,  as  explained  in  square  root,  is  also 
applicable  to  cube  root. 


VADE  MECUM.  W 


APPLICATION  OF  SQUARE  AND  CUBE 

ROOT,  WITH  MISCELLANEOUS 

MATTER. 

To  find  the  area  of  a  soalene  triangle. 

Rule.  From  the  half  sum  of  the  three  sides, 
take  the  three  sides  severally,  and  extract  the  square 
root  of  the  product  of  the  three  remainders  and 
half  sum. 

1.  Let  AB=5 
Let  B  C=6 

Let  AC =7 


5+6+7-4-2=9.^^ 


&— 5=4,  9—6=3,  9—7=2. 

4    X  3    X  2X9=^216=14.696+  Ans, 

2.  What  is  the  area   of   a   scalene  triangle, 
whose  sides  are  13,  14  and  15,  respectively  ? 

Ans.  84. 

3.  Given,  the  area  of  a  circle  1296,  to  find  th« 
side  of  a  square  equal  in  area. 

V1296=36.  ^^.  36. 

Given,  the  area  of  a  circle,  to  find  the  diameter. 

Hide.     Divide  the  area  by  \\,  and  extract  the 
square  root  of  the  quotient. 


58 


AKITHMiniCAL 


In  any  right  an- 
jjjled  triangle,  (see 
adjoining  figure, 
right-angled  at  C), 
when  one  leg  and 
the  liypotheniise  are 
given,  to  find  tlic 
otluir  leg. 

Kule.  Multi/>Jy 
the  suvi  by  the  dif- 
fercnce,  and   extract  the  square  root  of  tlie  j^roduct. 

When  the  two  legs  are  given,  to  find  the 
hypothennse. 

Rule.  To  double  their  product,  add  the  square 
of  their  difference,  and  extract  the  square  root  of 
the  sum. 

4.  What  is  the  hase  of  a  riglit-angled  triangle, 
whose   hypothcnnsc  is  10,  and  f)erpendicular  6  ? 

yr(J+6"xT0-^=8  Ans. 

5.  Given,  the  jterpendicnlar  3,  and  base  4,  to 
find  the  bypothenuse. 


^ax4xi-'+l-=5  Ans. 

Scholium  1.  Tiie  product  of  the  snm  and  dif- 
ference of  two  numbers,  is  equal  to  the  difference 
of  their  sfjuares. 

Scholium  2.  Double  the  product  of  two  num- 
bers, plus  the  sipiaic  of  their  difference,  is  equal 
to  the  sum  of  their  squares. 


YAM  mcasc. 


6.  A.  and  B.  start  from  the  same  point ;  A 
travels  due  east  24  miles,  and  B.  due  north  1>> 
miles  ;  how  far  are  they  apart  ?  Ans.  SO  M. 

7.  Two  men  start  from  the  same  point ;  one 
travels  15  miles  due  south,  the  other  25  milea 
south-east ;  how  far  are  they  apart  ?     Ans.  20  M. 

8.  What  is  the  mean  proportional  between  4 
and  9  ?  

^9x4=6^^5. 

9.  What  is  the  area  of  a  parallelogram,  whose 
diagonal  is  50,  and  the  sides  are  as  3  to  4  ? 

32+42=25  :  502  ..  3>^4  .  []=1200  Ans. 

10.  What  are  the  sides  of  a  parallelogram, 
♦vhose  area  is  1200,  and  proportional  of  sides  as 
3  to  4  ?  Ans.  30  and  40. 


3X4:  1200  ::oa.  n.^gr,,}. 
3X4:  1200  ::-i»  :  LJ=' 


^9UU=30. 
1^<^0.  ^1600=40. 

11.  Given,  the  dimensions  of  a  plank,  20  feet 
long,  18  inches  wide  at  one  end,  and  6  at  the 
other,  to  find  how  long  the  smaller  end  must  be 
to  contain  half  the  number  of  square  inches  in 
the  plank  ? 

Solution.  Draw 
A  B  D  C,  and  pro- 
duce A  B  and  C  D 
to  meet  in  O,  mak- 
ing the  triangle  A  0  C  30  feet  in  length,  con- 
Uining  22^  square  feet ;  the  triangle  B  0  D,  10 


60  ARITHMETICAL 

feet  in  length,  containing  2^-  square  feet.  Thei 
in  tlie  triangle  A  0  C,  we  only  have  to  obtain  a 
distance  from  O,  sufficient  to  contain  half  th« 
plank  A  B  D  C,  plus  2|  feet. 

22L— 2V==20,  No.  of  sq.  ft.  in  the  plank. 

20-^-2=10,  half  the  No.  of  sq.  ft.  in  the  plank. 

104-2L=12L. 

221  :  12L  ::  30^  :  []=500. 

^500=22.36— 10=12.36,  length  of  small  end. 

Note.     Areas  are  to  each  other  as  the  squarei 
of  their  similar  sides. 

12.  ^1  Rev.  J.  P.  MuRREL. 
How  long  a  line,  do  you  suppose, 
Would  just  the  amount  of  land  inclose, 
That  one  can  see  on  level  ground. 
Just  from  the  cent'r,  by  turning  round  j 
The  eye  about  six  feet  in  height, 
And  nothing  to  obstruct  the  sight  ? 

^ution  :  Ans.  18|  M. 

6-r'2«=V3+6=3IlX2=6DxV=18|  circumf. 

13.  If  an  eye  be  elevated  24  feet,  how  far,  on 
level  ground,  can  an  object  be  seen  ? 

24-2=12.         ^24+12=6  M.  Ans, 

14.  If  an  object  is  seen  9  miles,  how  high  is 
the  eye  elevated  ?  Ans.  54  ft. 

92x1=54  ft. 

15  By  Rev.  J.  P.  Murrel. 

Which  will  inclose  the  most  ground, 
A  fence  made  square,  or  one  made  round, 
Two  pannels  to  each  rod  of  land. 
Ten  rails  in  eachj  we  understand ; 


VADE    MECUM. 


61 


And  ev'ry  rail  in  each  suppose 
To  just  an  acre  of  land  inclose. 
The  next  thing  is  to  toll  exact, 
How  many  acres  in  each  tract? 


J        j  1024000,  No.  acres  in  sq*re. 
-^^^-  I  804571f,  '*      *'        circle. 

j-i^^=area  of  1  rod  square,  wliich 

takes  80  rails  to  fence  it. 

i 

j    ,-1^:80::      80      :  [1=1024000. 
Otaterrmus.  -j^^     :  11  ::1024000:  [J=804571f. 

Note.      One    side    of   the   square   equals   the 
diameter  of  the  circle. 


16.    In  what  time   will 
triple  itself,  at  5  per  cent  ? 

3— IXlOO-f 


any  sum   of  money 
Ans.  40  yrs. 

•5=40. 


k 


17.  In  what  time  will  any  sum  of  money 
double  itself,  at  6  per  cent  ?  Ans.  16|  yrs. 

2— lXl00-r-6=16|. 

18.  If  I  pay  850  apiece  for  7  mules,  and  the 
■arae  amount  for  horses  at  $70  apiece,  and  sell 
them  at  an  average  of  $60  apiece  ;  do  I  gain  or 
lose  ?  Ans.  $20  gain. 

19.  If  I  purchase  a  number  of  pears  at  3  cts. 
apiece,  and  pay  the  same  amount  for  oranges  at 
6  cents  apiece,  and  sr-U  them  all  i\i  an  average  of 
4  cents,  what  do  I  gain  or  lose  ?        Ans.  $0.00. 


62  ARITHMETICAL 

2yj.     A  horse  in  the  midst  of  a  meadow,  suppose, 
Made  fast  to  a  stake  by  a  cord  from  his  nose, 
How  long  must  this  cord  be,  that  feeding  all  'round, 
Permits  him  to  graze  just  two  acres  of  ground? 

Ann.  10.0904+. 
160x2-i-Vi=407.272727. 

7407.272727=20.1809=diameter  of  circle. 
Take  half  for  radius,  or  cord. 

21.  A  snail,  climbing  a  pole  20  feet  high, 
ascends  8  feet  per  day,  and  falls  back  4  feet  at 
night;  how  many  days  will  it  take  him  to  reach 
the  top  ?  Ans.  4  days. 

Statement— 1  :  4  ::  []  :  20—4=4. 

N^ote.  When  an  example  and  the  solution  are 
given,  the  solution  is  applicable  to  all  similar 
examples. 

22.  A.  can  do  a  piece  of  work  in  4  days,  B. 
in  6  days  ;  in  what  time  can  they  both  do  it 
working  together  ?  Ans.  2|  days. 

4x6-h4+6=2|. 

23.  As  I  was  beating  on  the  forest  ground, 
Up  starts  a  hare  before  my  two  grey  hounds, 
The  dogs  being  light  of  foot,  did  fairly  rua 
Unto  her  fifteen  rods,  just  twenty-one  ; 
The  distance  that  she  started  up  before 

Was  four-score,  sixteen  rods,  just,  and  no  more. 

Now,  I  would  have  you  unto  me  declare, 

How  far  they  ran  before  they  caught  the  hare  ? 

D        96       H  Ans.  336  rmls. 


21— 15  :  96  ::  21  :  [J. =336. 


VADE    MECUM.  68 

24.  The  hour  and  minute  hands  are  exactly 
together  at  12  o'clock;  at  what  time  will  they 
next  be  together  ? 

11  :  12  ::  1  :  [].=lh.  5min.  27f\sec. 

25.  The  hands  of  a  clock  are  together  between 
5  and  6  ;  what  is  the  time  ? 

11  :  12  ::  5  :  []=5h.  27min.  16yVsec. 

26.  The  hour  and  minnte  hands  are  directly 
opposite^he  minute  hand  between  4  and  5,  the 
hour  between  10  and  11  ;  what  is  the  time  ? 

Ans.  lOh.  21m.  49J,-sec. 

^oie.  It  is  the  same  time  past  10  as  it  would 
be  past  4,  were  the  hour  and  minute  hands  to- 
gether between  4  and  5. 

27.  The  time  past  noon  is  equal  to  ^  of  the 
time  past  midnight ;  what  is  the  time  ? 

Ans.  6  o'clock. 

Denominator — numerator  :  numerator  ::  12  :  []. 

2.  The  time  past  noon  is  equal  to  ^  of  the 
time  till  midnight ;  what  is  the  time  ? 

Ans.  3  o'clock. 

Denominator-|-numorator  :  numerator  ::  12  :  []. 

2o.     When  first  the  marriage-knot  was  tied. 

Between  my  ■wife  and  me, 
Jly  age  did  hcr-i  as  far  exceed 

As  three  times  three  does  three. 
But  after  ten.  and  half  ten  years, 

We  man  and  wife  had  been, 
Her  age  came  up  as  near  to  mine, 

As  e'ght  is  to  sixteen. 


64  ARITHMETICAL 

Now,  tyro,  skilled  in  numbers,  say, 
What  were  our  ages  on  wedding-day  ? 

_,.    .       j  3-r-3=l.     1x15=15.    - 
Solution.  I  9_^3^3^     3x15=45. 

29.  A  father  gave  to  his  son  f  of  his  whole 
estate ;  to  his  daughter  |  of  the  remainder,  and 
the  remaining  part  to  his  widow.  The  son  re- 
ceived ^75  more  than  the  daughter.  Required, 
the  share  of  each. 

C  8450=son's  share. 
Ans.  i  ^375=daughter's. 
(  ^225=widow's. 

80.  If  a  beam,  10  ft.  long,  5  in.  wide,  and  2 
in.  deep,  will  bear  up  100  lbs.,  how  many  lbs. 
will  another  support,  that  is  15  ft.  long,  6  in. 
deep,  and  3  in.  wide,  the  support  being  at  the 
end.  Ans.  360  lbs. 


2) 

6 

4 

—10  :  100  : 

:  6 

5 

3 

■15  :  [J.        ?  X^ 


6 
3 
1'0J3  2O 


360 

To  find  the  least  common  multiple  of  fractions. 

Hule.  Find  the  least  common  multipW^  of  tli% 
numerators,  and  divide  it  by  the  greatest  common 
divisor  of  the  denominators. 

*  The  process  is  the  same  as  in  Art.  13 


VADE    MEC'UM.  65 

31.  Find  the  least  common  multiple  of  |,  f , 
and  |.  Ans.  12. 

The  least  common  multiple  is  12. 
The  greatest  common  divisor  is  1. 
Thus  12-^1=12. 

32.  By  Ret.  J.  P.  Murrel. 
Suppose  two  walls  erect  should  stand, 
Across  a  street  on  either  hand; 
Now,  if  a  pole  should  stand  upright, 
Close  to  the  one  of  the  same  height, 
If  foot  be  drawn  twelve  feet  in  street. 
And  top  slips  down  about  two  feet  j 
Then  if  the  top  be  turned  to  fall, 

So  as  to  strike  the  other  wall 

Below  its  summit  just  six  feet  — 

What  are  their  heights,  and  width  of  street  ? 

.        J       37  ft.  height  of  walls. 
^"*-    (   82.2  ft.  width  of  street. 
Sohction. 
12^-|-2^-r-double  the  distance  the  pole  slipjped 
down=37  ft.,  height. 

37—6+37x37—31=408. 

V'408=20.2(nearly)+12=32.2,  width. 

33.  A  man  has  60  lbs.  of  wool,  worth  25  cts. 
per  lb.,  which  he  wishes  carded.  The  carder  has 
5  cents  per  lb.  for  carding,  and  takes  the  toll  in 
wool  before  carding  it ;  how  many  lbs.  must  he 
take  ?  Ans.  10  lbs. 

60x5-^-25+5=10. 

Falling  Bodies. 

Rule.     Multiply  the  square  of  the  timt  hy  16|*j 

feet,   {the  distance  a  body  will  fail  in,  ike  first 
second) . 


66  ARITHMETICAL 

34.  To  what  height  must  a  stone  be  raised  to 
require  it  4  seconds  to  reach  the  ground  ? 

Ans.  25Ti-  ft, 
4^X16  Jo=257i-. 

35.  What  is  the  depth  of  a  well,  to  the  bot- 
tom of  which  a  stone  would  be  3  seconds  in 
falling  ?  Ans.  144|  ft. 

36.  How  high  above  the  earth's  surface  must 
a  body  be  raised  to  lose  |  of  its  weight  ? 

Ans.  898.97. 

Rule.  Denominator — numerator  :  denomina- 
tor ::  square  of  the  earth's  semi-diameter  r  to 
the  square  of  the  distance  from  the  center  of  tht 
earth. 


3—1  :  3  ::  4000^  :  724000000=4898.97—4000 
=898.97  miles. 

37.  By  Rev.  J.  P.  Murrel. 
If  a  man  o'er  head,  in  a  balloon, 
Should  fire  a  level  gun  at  noon, 
How  high  is  he,  if  ball  and  sound, 

At  the  same  time,  should  strike  the  ground  ? 

Ans.  15i^-|-mile8. 
Note.     Sound  flies  1142  feet  per  second. 
11422-^16^2=^0.  of  feet. 

38.  Divide  100  apples  between  John  and  James, 
go  that  John  will  have  ^  more  than  James. 


VADE    MECUM.  67 


S9.  If  the  fourth  of  20  bo  3, 

What  will  the  fifth  of  30  be  ? 


40.  If  the  third  of  7  be  3, 

What  will  the  sixth  of  20  be? 


Ans.  3|. 


Ans.  4f . 

41 .  Two  pence  is  what  part  of  f  of  3  pence  ? 

Ans.  the  whole. 

42.  What  is  nothing,  twice  yourself,  and  50  ? 

A71S.  0.  W.  L. 

43.  Four-fifths  of  815  are  six-tenths  of  how 
many  thirds  of  821  ?  Ans.  2|. 

44.  Four-sevenths  of  42  are  f  of  how  many 
times  that  number  of  which  i-  of  9  is  f  ? 

Ans.  9. 

45.  If  A.,  B.,  C.  and  D.  start  from  the  same 
point  around  a  circular  island,  80  miles  in  cir- 
cumference, in  how  many  days  will  they  next  be 
together,  if  A.  travels  4  miles  per  day,  B.  8,  C. 
20,  and  D.  28?  Ans.  20. 

Mule.  Divide  the  distance  romid  the  island  by 
the  greatest  common  divisor  of  distances  traveled. 


46.  By  Rev.  J.  P.  Murrel. 

Suppose  a  horizontal  plane, 
On  which  did  stand  a  stalk  of  cane, 
The  height  of  which  I  took  quite  neat. 
And  found  to  be  one  hundred  feet. 
Soon  as  I  did  this  measure  take, 
A  blast  of  wind  this  cane  did  break. 


t$  ARITHMETICAL 

The  top  of  it  did  strike  the  ground 
Some  ten  yards  from  the  base,  I  found; 
One  end  of  which  did  still  remain, 
Just  where  the  wind  did  break  the  cane. 
Now,  can  you  all  these  measures  take, 
And  tell  how  high  the  cane  did  break  ? 

A71S.  45L  ft, 
1002  _302  _^ioO  X  2=45^. 

47.  Two  men,  A.  and  B.,  leave  Iowa  City. 

A.  travels  due  east,  39  miles,  to  Berlin  ;  B.  due 
soutli  to  St.  Francisville,  thence  to  Berlin,  and 
says  he  has  traveled  120  miles  since  he  left  Iowa 
City.  They  both  start  a  due  south  course,  and 
after  having  passed  the  latitude  of  St.  Francis- 
ville, travel  83  miles  to  Jefferson  Barracks. 
They  then  turn  a  due  west  course  to  a  point  ex- 
actly south  of  St.  Francisville.     They  here  part. 

B.  passes  directly  to  Iowa  City.  A.  continues 
his  course  due  west  to  Jefferson  City,  thence  to 
St.  Francisville,  antl  says,  since  he  parted  with 
B.  the  last  time,  he  has  traveled  100  miles. 
Returning  to  Jefferson  City,  he  inquires  how  far 
he  and  B.  are  separated.  Ans.  137.544-j-M. 

48.  What  is  the  length  of  a  trout,  whose  head 
is  3  inches  long,  his  tail  as  long  as  his  head  and 
I  of  his  body,  and  his  body  as  long  as  his  head 
and  tail  together  ?  Ans.  20  in. 

Head.  I  Body.  I         Tail. 


3      i     3      I      3     I     4     i      4     I     3 

49.  Edwin  bought  5  pears  for  5  cents  ;  Charles 
bought  3  for  3  cents  ;  being  afterwards  joined  by 
James,  the  three  made  a  meal  of  the  8  pears.  On 
leaving,   James    pays  them   8   cents,   of  which 


VADE    MECDM. 


69 


Charles  claims  3  cents,  as  he  furnished  3  pears. 
How,  in  equity,  should  the  8  cents  be  divided  ? 

J        j   Charles,  1  ct. 

^''^-    (  Edwin,    7  cts. 

50.  By  Rev.  J.  P.  Mukrei,. 

In  partnership,  wo  understand, 
Two  brothers  bought  a  piece  of  land, 
Two  hundred  acres  when  survey'd, 
And  each  four  hundred  dolhirs  paid. 
One  end  being  richer  than  the  other, 
The  elder  said  to  the  younger  brother, 
I'll  take  the  end  that 's  cot  so  poor. 
And  pay  a  half  a  dollar  more 
Per  acre,  if  you  will  agree, 
To  let  that  end  belong  to  me. 
To  which  the  younger  thus  replied, 
I  will,  if  you  '11  the  land  divide. 
T  will,  he  said,  and  at  it  went ; 
Eut,  after  ho  some  days  had  spent, 
And  found  it  did  his  soul  perplex. 
He  called  on  his  surveyor  next, 
Who  labored  hard,  but  could  not  quite 
Make  land  and  money  come  out  right ; 
Then  threw  it  down  with  grief  and  pain. 
Declaring  he'd  ne'er  try  again. 
Since  that,  this  sum  has  traveled  round, 
To  see  if  any  could  be  found 
Who  could  this  piece  of  land  divide. 
As  elder  brother  did  decide  ; 
Likewise  how  much  each  man  must  pay 
Per  acre,  in  his  own  survey. 
Now,  reader,  as  it's  come  to  you, 
Take  hold,  and  see  what  you  can  do ! 

{Elder  brother's,  93J4-^cres. 
Younger  brother's,  106^  acres,  nearly. 
Price  of  elder  brother's,  84.26.5+. 
Price  of  younger  brother's,  $3.76.5-j-. 

For  the  solution  of  the  above : 
Hule.     Find  the  cost  of  the  whole  number  of 
acres,  at  the  difference  between  the  j^rices  per  acrt, 


70 


ARITHMETICAL 


which  subtract  from  the  amount  paid  for  the  whole 
land ;  to  the  square  of  the  remainder,  add  the 
product  obtained  by  multiplying  the  cost  of  the 
whole  number  of  acres  {at  the  difference  between 
the  prices  per  acre),  by  four  times  the  whole  sum 
paid  by  him  who  paid  least  per  acre  ;  extract  tlie 
square  root  of  the  sum  ;  to  the  result  add  the 
remainder  that  was  squared,  and  divide  the  su7/i 
by  twice  the  whole  number  of  acres,  for  the  price 
per  acre  paid  by  him  who  paid  least  per  acre. 
Having  this,  other  requirements  of  the  question 
are  easily  found. 

51.  V>y  Rev.  J.  P.  Mubrel. 

If  round  a  point  two  wheels  you  start, 
By  axlo  kept  five  feet  apart, 
The  height  of  inner  wheel  complete, 
Supposed  to  be  about  three  feet — 
To  form  a  circle  would  require 
The  outer  wheel  a  little  higher  ; 
How  much  higher  would  you  suppose, 
That  inner  might  an  acre  inclose  ? 

Ans.  .1274+ft. 
117.728  ft.=radius  of  one  acre. 
117.728  :  117.728+5  ::  3  :  [].    3,1274+ft. 
height  of  larger  wheel. 

52.  By  Rev.  J.  P.  Murkel. 
Suppose  a  cart  drawn  once  around 
A  level  circular  piece  of  ground, 
Diameter  of  wheels  to  be, 

In  inches,  each  just  sixty-three. 

Now,  if  the  axle  of  this  cart 

Should  keep  the  wheels  five  feet  apart, 

IIow  many  times  will  one  turn  round 

More  than  the  other,  once  round  this  ground, 

If  inner  track  should  just  inclose. 

Five  thousand  acres  we'll  suppose  ? 

Ans.  l\\. 


VADE    MECUM. 


71 


5X   2xV=^^^^''^^"^^''^  ^^  peripherics. 
63-r-12xV=^^''^""^'^*^''^''^^'^  of  Avheels. 

2  2  o_i_3^  1  L2. 

7         •      2    ■^2  1* 

Note.  The  No.  of  acres  has  nothing  to  do  with 
ihe  solution. 

53.  How  wide  must  a  walk  he  around  a  rec- 
tangular garden,  24  yds.  long,  and  16  yds.  wide, 
to  contain  as  much  land  as  the  garden  ? 

c  1  ^.  Ans.  4  yds. 

Solution.  •' 

24x16^-4=06. 

24+16-^4=10.     10^  +96= ^196=14. 

14—10=4,  width  of  walk. 


54.  By  Rev.  J.  P.  Murbkl. 

Four  men,  A.,  B.,  C.  and  D., 

In  partnership  did  buy 
A  grindstone,  which  they  did  agree 

To  grind  away,  all  bat  the  eye. 
How  deep  in  radius  must  each  grind, 

To  have  an  equal  share  ; 
Diameter  fivo  feet  thoy  find, 

And  eye  three  inches  square  ? 


4.00+in.  1st  man. 

4.7o+in.  2d  man. 

6.15+in.  3d  man. 

12.99+in.4th  man. 


72  ARITHMETICAL 

602  xi-i-=2828  4  =areca  of  stone. 
32  _j_32  y^  Li=i4  1^  =area  of  circle  about  the  eye. 

4)2814  Y  =area  to  be  ground  away. 
703V|-==area  of  one  man's  share. 
J  28284— 703LfIZI=52.+.  60—52^-2=4 
■=radius  of  first  man's  share. 

2^ot€.  For  the  other  depths  of  radii  proceed 
in  the  same  manner. 

55.  Bought  2  watches  for  820  each,  and  sold 
one  at  25  per  cent,  gain,  the  other  at  20  per  cent, 
loss  ;  did  I  gain  or  lose  ?  Ans.  $1  gain. 

56.  When  5  pears  are  worth  7  peaches,  and 
12  peaches  15  apples,  and  21  apples  24  damsons, 
how  many  pears  can  I  have  for  36  damsons  ? 

Ans.  18  pears. 


$  n 


fX 


^  18 


18  pears. 


57^  By  Ret.  J.  P.  Murrel. 

How  large  a  field  would  be  required. 
Inclosed  by  fence  both  staked  and  ridered ; 
Two  pannels  to  each  rod  of  land, 
Ten  rails  in  each  we  understand, 
So  that  the  fence  may  just  inclose 
As  many  acres,  we'll  suppose. 
As  rails,  and  stakes,  and  riders  there, 
The  field  itself  to  be  a  square  ? 

Ans.  1730560  acres. 


VADE    MECUM.  78 

53^  By  Rev.  N.  C.  DeWitt. 

A  wealthy  maa  a  daughter  had, 
A  son  likewise  —  a  sprightly  lad  — 
To  whom  he  gave  a  piece  of  land, 
Which  was  a  square,  we  understand ; 
On  ev'ry  side  the  distance  found 
Was  just  four  hundred  rods  of  ground. 
The  next  thing  is  the  daughter's  share, 
Which  must  be  round,  and  not  a  eqnare  ; 
How  long  a  line,  do  you  suppose, 
Would  just  the  daughter's  land  inclose, 
And  she  the  same  amount  obtain 
That's  in  her  brother's  largo  domain  ? 

Ans.  1418  roda. 
Solution:     400x3.545=1418.000. 


59,  By   KeV.  J.  P.  MUKREL. 

A  youth  who  lived  a  single  life, 
Set  out  at  last  to  hunt  a  wife, 
So  to  a  house  he  did  repair, 
To  see  if  he  could  find  one  there. 
Directly  after  he  stepped  in. 

With  Miss a  courtship  did  begin, 

When  she  embraced  the  chance  to  find 
What  were  the  powers  of  his  mind. 
"  My  brother  and  myself,"  she  said, 
"Were  all  the  heirs  my  father  had; 
And  now,  kind  sir,  please  understand, 
We  both  were  in  a  foreign  land." 
Then  with  a  plaintive  voice  she  sighed, 
"  Pa  heard  that  one  of  us  had  died." 
Before  she  closed  she  added  still, 
"  While  we  were  there  Pa  made  his  will, 
Which  did  for  Ma  and  me  provide, 
If  I  had  lived  and  brother  died  : 
Two-thirds  of  his  estate  should  be 
Secured  to  Ma,  one-third  to  me. 
Had  brother  lived  and  I  had  died, 
For  them  the  will  did  thus  provide : 
Two-thirds  were  given  to  my  brother, 
And  only  one-third  left  to  mother. 
Now,  as  we  both  are  living  still.  y. 

And  must  be  governed  by  tlie  will, 

6 


Ans. 


74  ARITHMETICAL. 

Which  does  my  mother's  share  express, 
About  three  hundred  dollars  less 
Than  it  would  be  if  I  had  died, 
And  they  should  by  "the  will  abide. 
And  now,  kind  sir,  please  calculate. 
What  is  the  sum  of  Pa's  estate ; 
Likewise  how  much  each  share  will  be, 
According  to  the  will's  decree  ? 
Unless  you  do  all  these  decide, 
Vl\  not  consent  to  be  your  bride,  {a^ire.) 

Mother's,         61800. 

Daughter's,      $  900. 

Brother's,        ^SGOO. 
Solution:  I    Whole  estate, 86300. 

1=J>. 
2=M. 
4=B. 

7-r-3=2^,  had  daughter  died. 
2  ,  all  being  alive. 

T:300::2:[]=:$1800M's. 


00.  By  Ret.  S.  H.  Hodges. 

A  neighbor  asked  me  for  the  time, 
And  as  I  love  to  speak  in  rhyme, 
I  told  him  it  was  after  ten, 
And  both  the  hands  together  then. 
Tray  tell  me  now  the  time  precise, 
By  any  means  you  can  devise. 


Ans.  lOh.  54min.  82/|Sev 


61.  By  Rev.  S.  H.  Hodges. 

If  three-elevenths  of  the  age 

Of  one  who  is  a  noble  sage. 

By  twenty-eight  should  1^  increased, 

And  one  year  from  the  last  released. 


VADE    MECUM.  7* 

And  one-eleventh  of  a  year 
To  the  remainder  added  here, 
And  of  a  year  elevenths  three 
Be  now  subtracted,  yoa  will  sre,- 
You'll  have  one  half  the  sage's  yeara ; 
Now,  what  is  all  his  life  of  cares  ? 

A71S.  118  years. 

62.  By  E.  P.  TnoMPSOx. 

Suppose  a  mirror  sixteen  inches  wide, 
In  inches  long  precisely  twenty-four, 
The  frame  of  which  the  owner  does  decide, 
In  superfico  must  equal  it — no  mnre. 
How  wide  a  frame,  pray  unto  me  declare, 
That  each  shall  equal  be,  in  inches  square  ? 

Arts.  4  inches. 


63.  %  ^^^-  J-  P-  ^Iphrel. 

A  youth  who  lived  a  lonely  life. 

Concluded  he  must  have  a  wife  ; 

He  sought  a  fair  one  for  his  bride. 

Whose  father  just  before  had  died. 

The  fair  one's  heart  and  hand  were  gained, 

But  something  to  be  done  remained — 

The  maiden's  mother  must  consent, 

Before  the  matter  further  went. 

On  being  asked,  the  mother  said, 

"Why  do  you  bother  thus  ray  head  ? — 

To  you,  sir,  it  is  clearly  known 

That  I  have  just  five  daughters  grown. 

Their  fathers  will  says,  ^  my  first  four 

Must  have  ticdve  thousand,  and  no  more; 

To  my  lant  four  I  do  declare 

Eleven  tJious'and  is  their  share  ; 

To  my  last  three  and  first  I  give 

Ten  thoH'^nnd  dollars,  if  they  live  ; 

As  to  my  last,  and  my  frst  three, 

Nine  thousand  shall  their  portion  be  ; 

To  all,  except  the  third  alono, 

I  give  ci'/ht  thousand — now.  I'm  done.* 

Now,"  said  the  mother,  "if  you  tell 

The  third  one's  part,  I'll  give  you  Nell." 

Ans.  S4500. 


78 


ARITHMETICAL 


Solution :     12000=A.B.C.D. 

11000=B.C.D.E. 

10000=A.C.D.E. 

9000=A.B.C.E. 

800Q==A.E.D.E. 

4)50000=four  times  A.B.C.D.E. 

12500=A.B.C.D.E. 

8000=A.B.D.E. 


I 


4500==C.  (or  Hell.) 


64.         Suppose  a  cone  to  stand  upright, 
Which  is  one  foot  esa«t  in  height, 
How  high,  'bove  base  must  a  line  be, 
That  will  divide  it  equally  ? 

Ans.  2.4  76  inches. 


^12^-^2=9.5  24.     12—9.5  24=2.4  76. 


65. 

By  Rev.  S.  H.  Hodges. 

A  farmer  has  a  square  of  fertile 

land. 
And  in  the  center  all  his  build- 
ings stand ; 
His  land  he  has  determined  to 

divide 
Among  twelve  sons,  as  ho  can 
best  decide. 

Ho  first  proceeds  to  draw  a  circle  round. 

With  area  as  broad  as  all  his  ground ; 

And  next  he  doth  proceed  to  draw  a  square 

Whose  angles  in  the  circle's  bound'ry  are ; 

He  then  four  radii  doth  draw  complete. 

Which  in  four  points  with  square  and  circle  meet ; 

Eaoh  radius  in  length  is  sixty  poles, 

And  this  amount  alone  the  sum  controla. 


(/  \ 

\\    3 

/y 

VADE    ATECCM.  77 

The  liaes  now  made  divide  the  farmer's  ground, 
So  all  the  sons  may  have  their  portions  round. 
Now,  tell  me  how  much  land  there  is  in  all, 
And  how  much  will  within  the  circlo  fall — 
How  much  within  the  inner  square  will  be; 
And  also  each  son's  portion  tell  to  me. 
But  bear  in  mind,  the  first  four  each  must  take 
One-fourth  the  inner  square,  his  part  to  make. 
Unto  the  second  four  the  segments  give, 
That  they  within  the  circle's  rim  may  live. 
As  to  the  last  four,  give  them  for  their  lots 
The  outside  sections  —  smallest,  richest  spots. 

In  tlie  wliole  track,  90  acres 
In  tlie  circle,  70  y 

In  tlie  inner  square,  45 
First  four  sons,  next  the  farmer's,  each,  11  \ 
Second  four,  in  the  segments,  have  each    6  f 
Last  four,  in  the  outside  sections,  each,    4|j 

66.  Three  men.  A,  B,  and  C,  being  employed 
to  perform  a  certain  piece  of  work  for  8105, 
A  and  B  are  supposed  to  do  j^  of  it,  A  and  0 
7—,  and  B  and  C  f .  They  are  paid  proportion- 
ally ;  please  divide  their  pay  for  them  as  it 
should  be. 

67.  A  stick  of  timber  30  feet  long,  of  uniform 
thickness  and  breadth,  is  to  be  lifted  by  3  men, 
one  at  one  end,  and  the  other  two  holding  a 
hand-spike  near  the  other.  How  far  from  the 
end  must  they  be  placed  that  each  of  the  three 
may  raise  an  equal  portion  of  the  stick  ? 

68.  A  man  6  feet  high  traveled  round  the 
earth.  How  much  further  did  his  head  go  than 
his  feet  ? 


78 


AKITHMETICAL 


Am. 


By  Ret.  J.  P.  Muekel. 

There  is  a  Quaker,  Tre  understand, 
"WTio  for  three  sons  laid  off  his  land. 
And  made  three  equal  circles  meet, 
So  as  to  bound  an  acre  neat. 
Just  in  the  center  of  that  acre, 
Is  found  the  dwelling  of  the  Quaker ; 
In  centers  of  the  circles  round, 
A  dwelling  for  each  son  is  found  : 
Now,  can  you  tell,  by  skill  and  art. 
How  many  rods  they  live  apart  ? 

j Distance  from  son  to  son,  63  rods. 


\ 


father  to  son,  36.372  rods. 


Solution : 


^160x6.20156=31.5(nearly)=S  b. 
31.5x^=63=8  S. 
31.5-^2=151 =a  b. 
7|:L::15|:[]=9.093=a  q. 

^3]i2_i5.752_,27.279=S  a. 
27.279-1  9.093=36.372=q  S. 


vaoe  mecdm.  79 

70.  I^'Y  E.  p.  Thompson. 

The  author  of  this  work  I  chanced  to  ask, 

"  Tell  me,  sir,  if  you  please,  how  old  you  are?" 
And  he  replied,  "  Let  mc  impose  this  task. 

Which,  when  performed,  it  will  my  years  declare: 
If  to  my  a;i;o  one-fourth  score  added  be, 

And  of  that  sum  the  square  root  you  extract, 
And  add  it  to  the  sum,  you  then  will  see 

Th.'-t  you  will  have  just  one  score,  ten,  exact." 
You,  who  with  numbers  do  your  thoughts  engage. 

Pray  tell  me,  if  you  can,  what  is  his  age.         Ana.  — , 

71.  What  number  is  that,  to  which,  if  you  add 
the  square  root,  the  sum  will  be  42  ?       Ans.  36. 

Ride.  To  the  sum  (42),  add  one-fourth,  and 
extract  the  square  root,  from  which  take  one-half 
for  tli£  square  root  of  the  number, 

72.  By  E.  p.  Thompson. 

"Pray,"  said  a  lover  to  his  "gal," 

*'  Will  you  not  be  my  bride  ? 
My  heart,  my  hand,  my  all,  are  thine. 

Whatever  may  betide." 
To  which,  that  she  might  test  his  skill. 

The  pretty  thing  replied  : 
**A  gent  wa^  wont  to  visit  where 

Three  sisters  did  reside; 
And  go,  ere  starting  there  one  day, 

Within  his  pockets  wide 
He  put  of  pears  a  number  there, 

And  did  them  thus  divide  : 
To  Kate  he  gave  one-hajf  he  had. 

And  half  a  one  beside  ; 
The  half  then  left,  and  half  a  pear, 

For  Mol  he  did  provide  ; 
Half  then  remaining,  plus  one-half, 

For  Puss  he  set  aside  ; 
He'd  then  one  left.     Now,  tell  me,  sir,** 

The  gentle  maiden  sighed. 
''How  many  pears  in  all  h^d  he, 

To  'mong  the  three  divide  ? 
My  answer  must,  shall  be  delayed, 

Till  this  you  do  decide.  Ant.  — 


80  VADE    MECUM. 


IlE(JOMMENDATIONS. 


I  HAVE  examined,  with  much  interest,  many  portaona  of 
Mr.  I.  N.  Wilcoxson's  "Arithmetical  Vade  Mecum,"  (in 
manuscript),  and  find  myself  decidedly  favorable  to  the  -vyork. 
It  will  doubtless  prove,  not  only  a  convenient,  but  also  a  very 
useful  companion,  to  all  calculators  into  whose  bands  it  may 
fall.  I  have  been  a  teacher  of  youth,  the  most  of  my  time, 
for  about  fourteen  years ;  during  which  time,  T  have  exam- 
ined and  used  a  considerable  number  of  Arithmetics ;  but, 
for  conciseness,  perspicuity,  and  practical  utility,  I  feel 
constrained  to  praise  the  "  Vade  Mecum "  more  than  all  of 
them.  The  young  authol  of  this  new  work  deserves  the 
thanks  and  patronage  of  many,  for  the  great  improvement  he 
has  made  in  the  philosophy  and  practice  of  Arithmetic.  I 
am  strikingly  impressed  with  the  idea  of  oar  author  —  by 
his  untiring  energy  and  iyitense  study  —  becoming  an  instruc- 
tor of  instructors  t  Let  every  instructor,  merchant,  mechanic, 
student,  and  farmer,  procure  a  copy  of  this  new  work,  study 
it,  and  apply  its  rules  in  practice,  before  he  suffers  himself  to 
epeak  or  think  against  it. 

S.  H.  HODGES. 

Barren  County,  Kentucky. 


The  following  is  from  the  pen  of  James  G.  Hardy,  Lieu- 
tenant Governor  of  Kentucky : 

Glasgow,  Ky.,  Sept.  29,  1855. 
I  have  briefly  examined  the  "Arithmetical  Vade  Meeum," 
prepared  by  Isaac  N.  Wilcoxson,  Esq.  The  principles  of 
*^  Cause  and  Effect,"  or  a  separating  the  order  of  producing 
from  the  parts  produced,  are  worthy  the  consideration  and 
patronage  of  the  public,  and  I  have  no  hesitation  in  saying 
Ihat  the  work  will  be  useful  to  business  men. 

JAS.  G.  HARDY. 


A  SUPPLEMENT 


ARITHMETICAL  VADE  MECUM: 

GIVING  AN  EXPLANATION  AND  SHOWING  THE 

APPLICATION    OP    A    NEW    DIAGRAM;    Bi' 

MEANS  OF  WHICH  WE  READILY  APPLY 

CAUSE  AND  EFFECT  TO  NUMERICAL 

CALCULATION. 


There  is  but  one  rule  by  which  every  arith- 
metical question  of  a  practical  nature  in  arith, 
metic  may  be  scientifically  and  philosophically 
analyzed,  stated,  and  solved ;  it  being  founded 
upon  an  axiom  in  natural  philosophy,  "  That 
equal  causes  produce  equal  effects,  and  that 
effects  are  always  proportionate  to  their  causes.'^ 
There  are  but  two  primal  principles  —  increase 
and  decrease — in  numerical  calculation  ;  and.by^ 
the  same  statement,  we  increase  or  decrease  ac- 
cording to  the  nature  of  the  question.  This 
may  be  learned  by  those  of  common  capacity^ 
who  are  attentive,  and  understand  the  first  prin- 
ciples of  calculation,  in  a  very  few  days.  There 
is  contained  in  the  following  pages  of  the  Sup- 
plement the  labor  and  experience  of  years, 
teaching  and  investigating,  excluiively,  the  new 
system. 

I, owe  many  thanks  to  Ed.  Porter  Thompson,, 
now  Principal  of  Rich  Grove  Seminary,  a  new 
and  flourishing  Institute  on  the  turnpike  seven 
/miles  north  of  Glasgow,  Kentucky,  in  a  delight- 
ful neiorhb0r?hood,  for  much  assistance  rendered r 

ISAAC    N.  WILCOXSON. 
AuJtJUST  24,  3.858. 


SUPPLEMENT 


FIRST  COUPLKT.  SFX'OND    COLPLET. 

1st  Proposition. 2nrl  Proposition^ 

, — [means.] V 

Cause  :  Effect  :  :  Cause  :  Effect. 

(Isi  teiTQ.)         (2ud  terra.)  {3rd  term.)  (4rh  term.) 

[extremes.] — ' 


Question.  What  is  a  diagram  ? — Ansicer.  An 
arithmetical  scheme. 

Q.  What  system  is  here  used  : — A.  Dia- 
gramic  system  of  applying  cause  and  eff^ect  to 
numerical  calculation. 

Q.  What  is  the  ndel 
A.  Rule  : 

Cause  :  (is  to)  ej^ect  :  :  (as)  cause  :  (is  to)  effeft. 

Note  — The  prereding  is  the  Ri  i.e  complete,  but  it  may 
be  read  with  black  terms,  as  follows  : 

Cause  :  (is  to)  effect  :  :  (as)  cause  :  [ — ]*  (is  t<» 
required)  f-ffecf. 

*  Is  the  blank,  and  shows  the  place  of  the  required  term, 
or  factor. 


Bi  ARITHMETICAL 

or, 
Cause  :  (ia  to)  effect  :  :  [ — ]  (as  required)  cause 
:  (is  to)  ejlect. 

or, 
Cause  :  [ — ]  (is  to  required)  e^ect  : :  (as)  cause 
:  (i»  to)  ejhct, 

or, 
[ — ]  (Required)  cause  i  (is  to)  effect  :  :  (as)  cause 
:  (is  to)  effect.     (See  Art.  29.) 

Note. — Sometimes  there  is  only  a  factor  of  a  term  blank, 
or  wanting  ;  but  ihi«  does  not  alter  the  rule,  or  statement, 
in  the  least  ;  it  would  be  as  the  following  example  in  all 
Uie  terms. 

Cause  :  (is  to)  eject :  :  (as)  cause  :  (is  to)  (  factor  eject. 

c  [ — ]  ^'yquired  fac. 

Q.  What  is  a  term  ? — A,  One  member  of  a 
proportion, 

Q.  How  many  terms   in  tho  diagram  ? — A, 

N'oTE. — There  may  be  a  multitude  of  factors  ;  there  may 
he  several  in  a  term,  as  for  example:  If  3  men  in  8  days 
Mrld  a  -wall  20  feet  lomg,  12  feet  high,  and  2  feet  thick ;  how 
uiHiiy  men  would  be  required  in  4  days  to  bnild  another  24 
itet  long,  10  feet  high,  and  3  feet  thick  ? 


STATEMENT. 

c. 

E. 

C. 

E. 

Men    3 
Days  8 

length  20     :  ; 
height  12 
thickoess  2 

(-) 
4 

:     24 

10 

5 

Here  3  and  8  are  factors  of  the  first  term ;  3x8=24  the 
term;  20,  12,  and  2  are  factors  of  the  second  term;  20xl2x 
2=480  the  term. 


i 


VADE    MBOUM.  BS 

Q.  What  aie  they  divided  into  ? — A.  Means 
and  extremes. 

(See  Art.  26  A.  V.  M.  for  answers  to  follovy- 
ing  questions.) 

Q.  What  is  ratio  1 

Q.  What  is  a  couplet  1 

Q.  What  terms  compared  constitute  the  firsl 
couplet  ? 

Q.  What  terms  compared  constitute  the  see" 
ond  couplet  ? 

Q.  What  are  the  means  ? 

Q.  What  are  the  extremes  ? 

-Q.  What  is  a  proportion  ? 

Q.  The  product  of  the  means  must  equal 
what  ?  (See  Art.  27.) 

Note.  —  If  4  yds.  of  cloth  cost  Sl2,  6  yd«.  will  cost  hovy 
much  ? 

STATEMENT. 

C,  E.  C.  E. 

Yds.  4     :     $12     ::     6     :      (—J 

After  the  statement  is  made,  we  consider  the  factors  s« 
abstract  numbers.  In  the  first  couplet  the  effect  is  3  times 
as  large  as  its  cause  ;  so  must  the  second  efiect  be  3  times 
S.S  large  as  its  cause — 6x3=18  second  effect.  Then  13x4= 
12x6.     The  same  reasoning  applies  to  all  the  terms. 

Q.  What  is  a  proposition'? — A,  A  statement 
in  terms,  one  clause  of  a  question  containing 
two  t&ims,  as  4  yds.  of  cloth  cost  $12. 


86  ARITHMETICAL 

Q.  How  many  propositions  in  every  aritli- 
metical  question  ? — A.  Two. 

Q.   What  are  they! — A.  First  and  second. 

Q.  What  kind  are  they  ? — A.  Complete  and 
incomplete. 

Note. — The  complete  proposition  is' a  irjodel,  anwxample, 
by  which  to  arrange  or  complete  the  incomplete  proposition. 
The  complete  governs  the  incomplete. 

Q.  Where  is  the  first  proposition  arranged  ? 
— A.  In  the  first  couplet. 

Q.  How  ? — A.  To  suit  convenience. 

Q.  If  time  is  given,  where  is  it  placed? — A. 
In  the  same  term  with  that  that  can  produce  ac- 
tion or  agency. 

Q.  Where  is  the  second  proposition  arranged? 
— A.  In  the  second  couplet. 

Q.  How? — A.  Similar  to  the  first. 

Q.  What  is  cause? — A.  Action  or  agency. 

Q.  What  is  effect? — A  That  which  is  ac- 
complished or  follows  action  or  agency. 

Q.  What  are  the  guides  in  stating  the  ques- 
tion?— A.  The  elements  or  denominations  in  the 
question.     (See  Art.  31.) 

Q.  There  must  be  the  same  number  of  ele- 
ments or  factors  where? — A.  In  similar  terms. 
(See  Art.  31.     N.  B.) 


I 


VADE     MECUftl.  87 

Q.  What  must  be  in  similar  terms? — A.  The 
same  number  of  elements  ;  also,  the  same  num- 
ber of  factors,  though  one  may  be  a  blank  fac- 
tor. 

Q.  ^Causes  must  always  be  how  ? — A.  Of  the 
same  or  similar  kind. 

Q.  And  must  be  reduced  to  what  before 
placed  on  the  line  ? — A.  To  the  same  name  or 
denomination. 

Q.  Where  is  the  ( — )  blank  placed  ? — A.  In 
the  place  of  the  required  term  or  factor  ;  or  in 
that  term  similar  to  that  in  which  is  placed  the 
same  or  similar  element  or  name  in  the  arrange- 
ment of  the  complete  proposition. 

Note. — The  observing  student  will  notice  that  there  is 
an  action  or  agency  that  passes  from  the  cause  to  the  effect 
— a  relationship  shown — which  will  enable  bim  to  readily 
separate  the  factors  of  the  cause  from  those  of  the  effect, 
The  philosophical  way  of  staling  the  question  is  to  place  the 
acting  term  as  cause  ;,  but,  to  find  the  trEe  result  or  answer, 
it  is  immaterial  which  is  placed  as  cause,  so  that  the  second 
is  like  the  first. 

Q.  If  the  first  or  fourth  term  is  blank,  which 
are  complete,  means  or  extremes  ?  and  where 
are  they  placed  ? — A.  The  means  are  complete, 
and  are  placed  upon  the  right  as  a  dividend. 

Q.  Which  are  incomplete  ?  and  where  are 
they  placed  ? — A.  The  extremes  are  incomplete, 
and  are  placed  upon  the  left  as  a  divisor. 

Q.  If  the  blank  is  in  either  the  2nd  or  3rd 


88  ARITHMETICAL 

term,  which  are  complete,  means  or  extremes? 
and  where  are  they  placed  ? — A,  Extremes  are 
complete  and  are  placed  upon  the  right  as  a 
dividend. 

Q.  Which  are  incomplete?  and  where  are 
they  placed  ] — A.  The  means  are  incomplete, 
and  are  placed  upon  the  left  as  a  divisor. 

Q.  Then  what  is  involved  to  find  the  true 
result  or  answer  to  the  question  ? — A.  Nothing 
but  simple  multiplication  and  division. 

Q.  What  is  cancellation  used  for? — A.  To 
shorten  the  work  of  multiplication  and  division. 


INTEREST. 
(See  Art.  36  A.  V.  M.) 


PERCENTAGE. 
Cause       :       effect         :  :      cause     :     ej^ect. 
100  :  100-\-gain  "^  ct.  :  :  cost  price  :  se.i  g  price- 
or, 
100— loss  ^  ct. 


BARTER. 


Cause         :     effect 


lit  commodity  :  1  [ajnH)  :  :  2d  comm^y  :  i  (amH) 


Its  jpTice 


cause      :    effect. 


its  price. 


VADK     MKCUM. 

DISCOUNT. 


89 


Cause  :  effect  :  :     cause      :     effect. 

100    :    \00-{-interest  of    :  :  present    :    amount 
1  OOybr  given  time 


worth 


and  rate  ^  ct. 


to  he  dis- 
counted. 


EXAMPLE. 


What  is  the  present  worth  and  the  discount 
of  $110  for  1  year  and  8  months,  at  6  per  cent  ? 

Ans.  Pres.  $100.     Dis.  $10. 

C.  E.  C.  E. 

100     :     110     :  :     (— )     :     110=100  pres. 
ly.  87n.=20w.X6-M2=10+100=110. 


MENSURATION  TABLE. 
(See  Art.  46  A.  V.  M.) 

SQUARE  MEASURE. 

Cause       :        effect       :  :       cause      : 

Factors  of    :      U/iit  of    :  :     Length    : 
the  unit  of         measure             Width 
measure 

effect. 

No.  of 
units. 

90 

ARITHMETFCAL 

SOLID 

OR  CUBIC 

MEASURE 

Cause      : 
Factors  of    : 
the  unit  of 
measure 

effect       :  : 
Unit  of    :  : 
measure 

cause       : 

Length     : 

Width 

Depth 

effect. 
No.  of 
units. 

Note. — Tn  square  measure  put  iecgth  and  width,  atid  in 
solid  measure  put  length,  width,  and  depth,  under  the  sec- 
oud  cause,  and  reduce  the  first  cause  to  the  same  denomi- 
nation, then  place  it  on  the  line  for  solution. 


Q.  What  is  the  first  proposition  in  interest  ? 
- — A.  100,  1  year,  and  rate  per  cent. 

Q.  What  is  the  second  proposition  1 — A. 
Principal,  time,  and  interest, 

Q.  What  is  the  first  proposition  in  percent" 
age  1 — A.  100  and  its  amount. 

Q.  How  do  you  find  that  amount? — A.  By 
adding  the  gain  per  cent  to,  or  subtracting  the 
loss  per  cent  from,  100. 

Q.  What  is  the  second  proposition  in  per- 
centage 1 — A.  Cost  and  selling  price. 

Q.  What  is  the  first  proposition  iti  barter  ? — 
A.  First  commodity,  its  price,  and  1  (amount). 

Q.  What  is  the  second  proposition? — A.  Sec- 
ond commodity,  its  price,  and  1  (amount). 

Q.  What  is  the  first  proposition  in  discount? 
— ^.   100  and  its  amount. 


I 


VADE      MECUM. 


91 


Q.  How  is  that  amount  found? — A.  By  get- 
ting the  interest  on  100  for  given  time  and  rate 
per  cent  and  adding  it  to  the  100. 

Q.  What  is  the  second  proposition  in  dis- 
count?— A.  Present  worth  and  amount  tc  be 
discounted. 

Q.  What  is  the  first  proposition  in  measure- 
ment 1 — A,  The  unit  of  measure  and  its  factors. 

Q,.  What  is  the  second  proposition? — A.  The 
factors  of  the  thing  to  be  rneasuied,  and  the 
number  of  units  it  contains. 

Note. — By  the  foregoing  questions  the  investigator  csin 
dearly  see  that  there  is  only  the  one  rale  and  principle 
necessary  to  analyze,  state,  and  solve  all  questions  coming 
up  under  the  preceding  specified  rules.  It  may  also  be 
applied  to  any  practical  calculation  in  life. 


MISCELLANEOUS  QUESTIONS. 

1.  Add  together  \l\l  \\  and  i. 

3 
2 

2 
3 


6 

5 

30 

3 

2 

24 

4 

1 

9 

9 

i 

28 

12 

11 

33 

2 

1 

18 

36 

142= 

Least  com.  denom. 


2.  Add  together  \  \  I  \  li- 


=3U 

(See  A.  V.  M.) 

Ans.  2%. 


9J  ARITHMETICAL 

3.  From  j|  take  |.  Ans.  j|. 

4.  Multiply  together  ^i  ^i  |  ^  -|  ||  and  |. 

Ans.  15, 

5.  Multiply  together  |  |  |  |   i;  3  1|  and  If. 

Ans.  2|. 

6.  Divide  |  by  |,  ^w*  2. 

7.  Divide  i  by  |.  ^7^5.  |. 

Note. — Multiplying  by  a  fraction  decreases  the  multipli- 
cand ;  and  dividing  by  a  fraction  increases  the  divideEd. 

8.  Divide  4  by  |.  Ans.  12. 

9.  Divide  |  by  5.  Ans.  ji^ 

10.  Divide  |  of  4  by  j|  of  2,  and  multiply  the 
quotient  by  2|  of  |,  and  divide  by  3i  of  |. 

Ans.  3. 

11.  If  8  hats  cost  $40,  what  v/ill  6  hats  cost? 

Ans.  $30. 

12.  If  200  ft>  of  pork  cost  $10,  how  many  ib 
can  I  have  for  840  ?  Ans.  800  ft) 

13.  If  a  pole  7  feet  high,  at  noon  east  a  sha- 
dow 5  feet  long,  what  is  the  height  of  a  tree 
whose  shadow  measures  80  feet?     Ans.  112  ft. 

14.  If  3  men  in  10  days  build  20  rods  of 
fence,  how  long  a  time  should  be  allowed  5 
men  to  build  50  rods  ?  Ans.  15  days. 

15.  If  $200  in  8  months  gain  $10  interest, 
how  many  months  will  it  take  the  same  prin- 
cipal to  amount  to  $250  ?  Ans.  40m, 


VADE     MEUUM.  93 

16.  Suppose  1  woman  in  2J  days  sews  the 
seams  of  two  pair  of  pants,  each  of  which  aver- 
age 4 1  seams  30  inches  long  and  12  stitches  ta 
the  inch,  how  many  days  should  it  take  5  women 
to  make  12  pair  each  of  which  averages  4^ 
seams  48  incEes  long  and  15  stitches  to  the 
inch  ?  Ans.  5  days. 

17.  When  10  men  in  6  days  dig  a  trench 
40  feet  long,  5  feet  wide,  and  S  feet  deep,  that 
is  of  the  hardness  of  3,  how  many  days  ought 
it  to  take  80  men  to  dig  another  that  is  400  feet 
long,  8  feet  wide,  18  feet  deep,  and  of  the  hard- 
ness of  4  ?  Ans  36  days» 

"What  is  the  interest  of— 

18.  $235  for  30  days  at  12  per  ct. 

Ans.  2.35 

19.  $350  for  60  days  at  6  per  ct. 

Ans.  3.60 

20.  $700  for  90  days  at  4  per  ct. 

Ans.  7.0O 

21.  $360  for  2  days  at  6  per  ct. 

Ans.  12cts. 

22.  S120  for  ly.  4m.  20d.  at  6  per  ct. 

Ans.  10.00 

23.  What  principal  at  interest  for  8  months 
and  10  days,  at  6  per  ct.,  will  draw  $15  interest? 

Ans.  360  00 

24.  At  what  rate  per  ct.  will  $900,  in  1  year, 
1  month  and  10  days,  draw  $60  interest  ? 

Ans.  6  per  ct. 


94  ARITHMETICAL 

25.  In  what  time  will  $1000,   at  6  per  ct., 
<lra\v  $100  interest  ?  Ans.  20m. 

How  must  the  following  articles  be  sold,  that 

€OSt — 

26.  81  cts.  to  clear  50  per  ctn? 

Ans.  12 J  cts. 

27.  10  cts.  to  clear  50  per  ct.1 

Ans.  15  cts. 

28.  12  J  cts.  to  clear  50  per  ct.? 

Ans.  18|  cts. 

29.  80  cts.  to  clear  25  per  ct.? 

Ans.  $1.00 
SO.  60  cts  to  clear  33J  per  ct.] 

Ans.  80  cts. 

31.  10  cts.  to  clear  10  per  ct.] 

Ans.  1 1  cts. 

32.  $5  to  clear  20  per  ct.1  Ans.  $6. 

33.  H  to  lose  25  per  ct.1  Ans.  $3. 

34.  $1  to  lose  10  per  ct.t  Ans.  90  cts. 

35.  Pay  25  cts.  for  an  article,  and  sell  it  for 
30  cts.,  what  per  ct.  is  gained  ? 

Ans.  20  per  ct. 

36.  Pay  5  cts.,  and  sell  at  8i,  what  per  ct.  is 
gained  ]  Ans.  66|. 

37.  Pay  40  cts.,  and  sell  at  50,  what  per  ct. 
is  gained  1  Ans.  25. 

38.  Pay  I2l,  and  sell  at  10,  what  per  ct.  is 
lost?  '  Ans.  20. 


TADE     MECL'M.  95 

39.  Sold  a  horse  for  $75,  and  lost  25  per  ct., 
what  did  he  cost  1  Ans.  $100. 

40.  Sold  coffee  at  22  cts.  per  lb.  and  gained 
10  per  ct.,  what  did  it  cost  1  Ans.  20  cts. 

41.  Barter  S  lb  of  butter  at  12.}  cts.  per  ft> 
for  domestic  at  10  cts.  per  yard,  how  many 
yards  mnst  I  have  ?  Ans.  10  yds. 

What  is  the  present  worth  and  discount  of — 

42.  8303  for  Im.  15cl.  at  8  per  ct.] 

Ans.  P.  W.  S300.     D.  $3. 

43.  $166  for  ly.  and  8m.  at  6  per  ct.? 

Ans.  P.  W.  $15U.     D.  $15. 

44.  $i21.80  for  90  davs,  at  6  per  ct.? 

Ans.  P.  \V.  S120.     D.  $1.80. 

45.  $100  f  )r  1  year  at  6  per  ct.? 

Ans.  P.  W.  $941?.     D.  85||. 

46.  SI 00  for  one  year  at  10  per  ct.? 

Ans.  P.  W.  $90|f.     D.  9i. 

47.  A  note  of  $500  was  given,  January  I, 
1857,  and  indorsed  July  4,  1857,  by  3115.331; 
another  indorsement  made  December  25,  1857, 
of  $211. 33|,  what  was  due  March  1,  1S58-. 
and  if  interest  was  paid  oh  it  then,  what  is  due 
to-day,  interest  at  6  per  ct,? 

Due  March  1,  '58,  $202. 16|. 
Due  to-day,  $ 

Note. — Bank  Discount  is  simple  interest  calcalated  npon 
amount,  with  "  three  days  of  grace  "  add«d  to  the  time  : 
this  tak«n  from  the  amount  will  leave  the  principal. 


J 


^^    I. 


UC  SOUTHERN  REOO'.A,     :PCiD/c< 


llililillllllllilll „„ 

B     000  012  943     7 


